Methods and apparatuses for signaling with geometric constellations in a raleigh fading channel

ABSTRACT

Communication systems are described that use signal constellations, which have unequally spaced (i.e. ‘geometrically’ shaped) points. In many embodiments, the communication systems use specific geometric constellations that are capacity optimized at a specific SNR, over the Raleigh fading channel. In addition, ranges within which the constellation points of a capacity optimized constellation can be perturbed and are still likely to achieve a given percentage of the optimal capacity increase compared to a constellation that maximizes d min , are also described. Capacity measures that are used in the selection of the location of constellation points include, but are not limited to, parallel decode (PD) capacity and joint capacity.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present invention is a continuation of U.S. patent application Ser.No. 13/179,383 entitled “Methods and Apparatuses for Signaling WithGeometric Constellations in a Raleigh Fading Channel”, which is issuedas U.S. Pat. No. 9,191,148 on Nov. 17, 2015, to Barsoum et al., filedJul. 8, 2011, which application claims priority as aContinuation-In-Part to U.S. patent application Ser. No. 12/156,989entitled “Design Methodology and Method and Apparatus for Signaling withCapacity Optimized Constellation”, which issued as U.S. Pat. No.7,978,777 and claims priority to U.S. Provisional Application Ser. No.60/933,319 entitled “New Constellations for Communications Signaling:Design Methodology and Method and Apparatus for the New SignalingScheme” to Barsoum et al., filed Jun. 5, 2007. The present applicationalso claims priority to U.S. Provisional Application Ser. No. 61/362,649filed Jul. 8, 2010 entitled “Methods and Apparatuses for Signaling withGeometric Constellations in a Raleigh Fading Channel” to Barsoum et al.The disclosures of U.S. patent application Ser. Nos. 13/179,383,12/156,989 and U.S. Provisional Application No. 61/362,649 are expresslyincorporated by reference herein in their entirety.

STATEMENT OF FEDERALLY SPONSORED RESEARCH

This invention was made with Government support under contractNAS7-03001 awarded by NASA. The Government has certain rights in thisinvention.

BACKGROUND

The present invention generally relates to bandwidth and/or powerefficient digital transmission systems and more specifically to the useof unequally spaced constellations having increased capacity.

The term “constellation” is used to describe the possible symbols thatcan be transmitted by a typical digital communication system. A receiverattempts to detect the symbols that were transmitted by mapping areceived signal to the constellation. The minimum distance (d_(min))between constellation points is indicative of the capacity of aconstellation at high signal-to-noise ratios (SNRs). Therefore,constellations used in many communication systems are designed tomaximize d_(min). Increasing the dimensionality of a constellationallows larger minimum distance for constant constellation energy perdimension. Therefore, a number of multi-dimensional constellations withgood minimum distance properties have been designed.

Communication systems have a theoretical maximum capacity, which isknown as the Shannon limit. Many communication systems attempt to usecodes to be able to transmit at a rate that is closer to the capacity ofa communication channel. Significant coding gains have been achievedusing coding techniques such as turbo codes and LDPC codes. The codinggains achievable using any coding technique are limited by theconstellation of the communication system. The Shannon limit can bethought of as being based upon a theoretical constellation known as aGaussian distribution, which is an infinite constellation where symbolsat the center of the constellation are transmitted more frequently thansymbols at the edge of the constellation. Practical constellations arefinite and transmit symbols with equal likelihoods, and therefore havecapacities that are less than the Gaussian capacity. The capacity of aconstellation is thought to represent a limit on the gains that can beachieved using coding when using that constellation.

Prior attempts have been made to develop unequally spacedconstellations. For example, a system has been proposed that usesunequally spaced constellations that are optimized to minimize the errorrate of an uncoded system. Another proposed system uses a constellationwith equiprobable but unequally spaced symbols in an attempt to mimic aGaussian distribution.

Other approaches increase the dimensionality of a constellation orselect a new symbol to be transmitted taking into considerationpreviously transmitted symbols. However, these constellation were stilldesigned based on a minimum distance criteria.

SUMMARY OF THE INVENTION

Systems and methods are described for constructing a modulation suchthat the constrained capacity between a transmitter and a receiverapproaches the Gaussian channel capacity limit first described byShannon [ref Shannon 1948]. Traditional communications systems employmodulations that leave a significant gap to Shannon Gaussian capacity.The modulations of the present invention reduce, and in some cases,nearly eliminate this gap. The invention does not require speciallydesigned coding mechanisms that tend to transmit some points of amodulation more frequently than others but rather provides a method forlocating points (in a one or multiple dimensional space) in order tomaximize capacity between the input and output of a bit or symbol mapperand demapper respectively. Practical application of the method allowssystems to transmit data at a given rate for less power or to transmitdata at a higher rate for the same amount of power.

One embodiment of the invention includes a transmitter configured totransmit signals to a receiver via a communication channel, where thetransmitter, includes a coder configured to receive user bits and outputencoded bits at an expanded output encoded bit rate, a mapper configuredto map encoded bits to symbols in a symbol constellation, a modulatorconfigured to generate a signal for transmission via the communicationchannel using symbols generated by the mapper, where the receiver,includes a demodulator configured to demodulate the received signal viathe communication channel, a demapper configured to estimate likelihoodsfrom the demodulated signal, and a decoder that is configured toestimate decoded bits from the likelihoods generated by the demapper. Inaddition, the symbol constellation is a PAM-8 symbol constellationhaving constellation points within at least one of the ranges specifiedin FIGS. 25-47.

In a further embodiment, the code is a Turbo code. In anotherembodiment, the code is a LDPC code.

In a still further embodiment, the constellation provides an increase incapacity over the Raleigh fading channel at a predetermined SNR that isat least 5% of the gain in capacity achieved by a constellationoptimized for joint capacity at the predetermined SNR.

In still another embodiment, the constellation provides an increase incapacity over the Raleigh fading channel at a predetermined SNR that isat least 20% of the gain in capacity achieved by a constellationoptimized for joint capacity at the predetermined SNR.

In a yet further embodiment, the constellation provides an increase incapacity at a predetermined SNR over the Raleigh fading channel that isat least 50% of the gain in capacity achieved by a constellationoptimized for joint capacity at the predetermined SNR.

In yet another embodiment, the constellation provides an increase incapacity over the Raleigh fading channel at a predetermined SNR that isat least 90% of the gain in capacity achieved by a constellationoptimized for joint capacity at the predetermined SNR.

In another embodiment again, the constellation provides an increase incapacity over the Raleigh fading channel at a predetermined SNR that isat least 100% of the gain in capacity achieved by a constellationoptimized for joint capacity at the predetermined SNR.

In a further additional embodiment, the constellation provides anincrease in capacity over the Raleigh fading channel at a predeterminedSNR that is at least 5% of the gain in capacity achieved by aconstellation optimized for PD capacity at the predetermined SNR.

In another additional embodiment, the constellation provides an increasein capacity over the Raleigh fading channel at a predetermined SNR thatis at least 20% of the gain in capacity achieved by a constellationoptimized for PD capacity at the predetermined SNR.

In a still yet further embodiment, the constellation provides anincrease in capacity over the Raleigh fading channel at a predeterminedSNR that is at least 50% of the gain in capacity achieved by aconstellation optimized for PD capacity at the predetermined SNR.

In still yet another embodiment, the constellation provides an increasein capacity over the Raleigh fading channel at a predetermined SNR thatis at least 90% of the gain in capacity achieved by a constellationoptimized for PD capacity at the predetermined SNR.

In still another embodiment again, the constellation provides anincrease in capacity over the Raleigh fading channel at a predeterminedSNR that is at least 100% of the gain in capacity achieved by aconstellation optimized for PD capacity at the predetermined SNR.

A still further additional embodiment includes a transmitter configuredto transmit signals to a receiver via a communication channel, where thetransmitter, includes a coder configured to receive user bits and outputencoded bits at an expanded output encoded bit rate, a mapper configuredto map encoded bits to symbols in a symbol constellation, a modulatorconfigured to generate a signal for transmission via the communicationchannel using symbols generated by the mapper, where the receiver,includes a demodulator configured to demodulate the received signal viathe communication channel, a demapper configured to estimate likelihoodsfrom the demodulated signal, and a decoder that is configured toestimate decoded bits from the likelihoods generated by the demapper. Inaddition, the symbol constellation is a PAM-16 symbol constellationhaving constellation points within at least one of the ranges specifiedin FIGS. 48-71.

Still another additional embodiment includes a transmitter configured totransmit signals to a receiver via a communication channel, where thetransmitter, includes a coder configured to receive user bits and outputencoded bits at an expanded output encoded bit rate, a mapper configuredto map encoded bits to symbols in a symbol constellation, a modulatorconfigured to generate a signal for transmission via the communicationchannel using symbols generated by the mapper, where the receiver,includes a demodulator configured to demodulate the received signal viathe communication channel, a demapper configured to estimate likelihoodsfrom the demodulated signal, and a decoder that is configured toestimate decoded bits from the likelihoods generated by the demapper. Inaddition, the symbol constellation is a PAM-32 symbol constellationhaving constellation points within at least one of the ranges specifiedin FIGS. 72-95.

Another further embodiment includes a transmitter configured to transmitsignals to a receiver via a communication channel, where thetransmitter, includes a coder configured to receive user bits and outputencoded bits at an expanded output encoded bit rate, a mapper configuredto map encoded bits to symbols in a symbol constellation, a modulatorconfigured to generate a signal for transmission via the communicationchannel using symbols generated by the mapper, where the receiver,includes a demodulator configured to demodulate the received signal viathe communication channel, a demapper configured to estimate likelihoodsfrom the demodulated signal, and a decoder that is configured toestimate decoded bits from the likelihoods generated by the demapper. Inaddition, the symbol constellation is a N-Dimensional symbolconstellation, where the constellation points in at least one dimensionare within at least one of the ranges specified in FIGS. 25-95.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a conceptual illustration of a communication system inaccordance with an embodiment of the invention.

FIG. 2 is a conceptual illustration of a transmitter in accordance withan embodiment of the invention.

FIG. 3 is a conceptual illustration of a receiver in accordance with anembodiment of the invention.

FIG. 4a is a conceptual illustration of the joint capacity of a channel.

FIG. 4b is a conceptual illustration of the parallel decoding capacityof a channel.

FIG. 5 is a flow chart showing a process for obtaining a constellationoptimized for capacity for use in a communication system having a fixedcode rate and modulation scheme in accordance with an embodiment of theinvention.

FIG. 6a is a chart showing a comparison of Gaussian capacity and PDcapacity over the AWGN channel for traditional PAM-2,4,8,16,32.

FIG. 6b is a chart showing a comparison between Gaussian capacity andjoint capacity over the AWGN channel for traditional PAM-2,4,8,16,32.

FIG. 7 is a chart showing the SNR gap to Gaussian capacity for the PDcapacity and joint capacity over the AWGN channel of traditionalPAM-2,4,8,16,32 constellations.

FIG. 8a is a chart comparing the SNR gap to Gaussian capacity of the PDcapacity for traditional and optimized PAM-2,4,8,16,32 constellations,over the AWGN channel.

FIG. 8b is a chart comparing the SNR gap to Gaussian capacity of thejoint capacity for traditional and optimized PAM-2,4,8,16,32constellations, over the AWGN channel.

FIG. 9 is a chart showing Frame Error Rate performance of traditionaland PD capacity optimized PAM-32 constellations in simulations involvingseveral different length LDPC codes, over the AWGN channel.

FIGS. 10a-10d are locus plots showing the location of constellationpoints of a PAM-4 constellation optimized for PD capacity and jointcapacity over the AWGN channel versus user bit rate per dimension andversus SNR.

FIGS. 11a and 11b are design tables of PD capacity and joint capacityoptimized PAM-4 constellations for the AWGN channel in accordance withembodiments of the invention.

FIGS. 12a-12d are locus plots showing the location of constellationpoints of a PAM-8 constellation optimized for PD capacity and jointcapacity over the AWGN channel versus user bit rate per dimension andversus SNR.

FIGS. 13a and 13b are design tables of PD capacity and joint capacityoptimized PAM-8 constellations for the AWGN channel in accordance withembodiments of the invention.

FIGS. 14a-14d are locus plots showing the location of constellationpoints of a PAM-16 constellation optimized for PD capacity and jointcapacity over the AWGN channel versus user bit rate per dimension andversus SNR.

FIGS. 15a and 15b are design tables of PD capacity and joint capacityoptimized PAM-16 constellations for the AWGN channel in accordance withembodiments of the invention.

FIGS. 16a-16d are locus plots showing the location of constellationpoints of a PAM-32 constellation optimized for PD capacity and jointcapacity over the AWGN channel versus user bit rate per dimension andversus SNR.

FIGS. 17a and 17b are design tables of PD capacity and joint capacityoptimized PAM-32 constellations for the AWGN channel in accordance withembodiments of the invention.

FIG. 18 is a chart showing the SNR gap to Gaussian capacity fortraditional and capacity optimized PSK constellations over the AWGNchannel.

FIG. 19 is a chart showing the location of constellation points of PDcapacity optimized PSK-32 constellations over the AWGN channel.

FIG. 20 is a series of PSK-32 constellations optimized for PD capacityover the AWGN channel at different SNRs in accordance with embodimentsof the invention.

FIG. 21 illustrates a QAM-64 constructed from orthogonal Cartesianproduct of two PD optimized PAM-8 constellations in accordance with anembodiment of the invention.

FIGS. 22a and 22b are locus plots showing the location of constellationpoints of a PAM-4 constellation optimized for PD capacity over a Raleighfading channel versus user bit rate per dimension and versus SNR.

FIGS. 23a and 23b are locus plots showing the location of constellationpoints of a PAM-8 constellation optimized for PD capacity over a Raleighfading channel versus user bit rate per dimension and versus SNR.

FIGS. 24a and 24b are locus plots showing the location of constellationpoints of a PAM-16 constellation optimized for PD capacity over aRaleigh fading channel versus user bit rate per dimension and versusSNR.

FIGS. 25-27 are tables showing the performance of geometric PAM-8constellations optimized for Joint Capacity over a Raleigh fadingchannel at specific SNRs in accordance with embodiments of theinvention.

FIGS. 28-30 are tables listing the constellation points corresponding tothe geometric PAM-8 constellation designs optimized for Joint Capacityover a Raleigh fading channel at specific SNRs listed in FIGS. 25-27.

FIGS. 31-32 are tables showing maximum ranges for the geometric PAM-8constellation designs optimized for Joint Capacity over a Raleigh fadingchannel at specific SNRs listed in FIGS. 25-27.

FIGS. 33-37 are tables showing the performance of geometric PAM-8constellations optimized for PD Capacity over a Raleigh fading channelat specific SNRs in accordance with embodiments of the invention.

FIGS. 38-43 are tables listing the constellation points corresponding tothe geometric PAM-8 constellation designs optimized for PD Capacity overa Raleigh fading channel at specific SNRs listed in FIGS. 33-37.

FIGS. 44-47 are tables showing maximum ranges for the geometric PAM-8constellation designs optimized for PD Capacity over a Raleigh fadingchannel at specific SNRs listed in FIGS. 33-37.

FIGS. 48-49 are tables showing the performance of geometric PAM-16constellations optimized for Joint Capacity over a Raleigh fadingchannel at specific SNRs in accordance with embodiments of theinvention.

FIGS. 50-53 are tables listing the constellation points corresponding tothe geometric PAM-16 constellation designs optimized for Joint Capacityover a Raleigh fading channel at specific SNRs listed in FIGS. 48-49.

FIGS. 54-55 are tables showing maximum ranges for the geometric PAM-16constellation designs optimized for Joint Capacity over a Raleigh fadingchannel at specific SNRs listed in FIGS. 48-49.

FIGS. 56-59 are tables showing the performance of geometric PAM-16constellations optimized for PD Capacity over a fa Raleigh fadingchannel at specific SNRs in accordance with embodiments of theinvention.

FIGS. 60-67 are tables listing the constellation points corresponding tothe geometric PAM-16 constellation designs optimized for PD Capacityover a Raleigh fading channel at specific SNRs listed in FIGS. 56-59.

FIGS. 68-71 are tables showing maximum ranges for the geometric PAM-16constellation designs optimized for PD Capacity over a Raleigh fadingchannel at specific SNRs listed in FIGS. 56-59.

FIGS. 72-73 are tables showing the performance of geometric PAM-32constellations optimized for Joint Capacity over a Raleigh fadingchannel at specific SNRs in accordance with embodiments of theinvention.

FIGS. 74-81 are tables listing the constellation points corresponding tothe geometric PAM-32 constellation designs optimized for Joint Capacityover a Raleigh fading channel at specific SNRs listed in FIGS. 72-73.

FIGS. 82-83 are tables showing maximum ranges for the geometric PAM-32constellation designs optimized for Joint Capacity over a Raleigh fadingchannel at specific SNRs listed in FIGS. 72-73.

FIGS. 84-85 are tables showing the performance of geometric PAM-32constellations optimized for PD Capacity over a Raleigh fading channelat specific SNRs in accordance with embodiments of the invention.

FIGS. 86-93 are tables listing the constellation points corresponding tothe geometric PAM-32 constellation designs optimized for PD Capacityover a Raleigh fading channel at specific SNRs listed in FIGS. 84-85.

FIGS. 94-95 are tables showing maximum ranges for the geometric PAM-32constellation designs optimized for PD Capacity over a Raleigh fadingchannel at specific SNRs listed in FIGS. 84-85.

DETAILED DESCRIPTION OF THE INVENTION

Turning now to the detailed description of the invention, communicationsystems in accordance with embodiments of the invention are describedthat use signal constellations, which have unequally spaced (i.e.‘geometrically’ shaped) points. In many embodiments, the communicationsystems use specific geometric constellations that are capacityoptimized at a specific SNR. In addition, ranges within which theconstellation points of a capacity optimized constellation can beperturbed and are still likely to achieve a given percentage of theoptimal capacity increase compared to a constellation that maximizesd_(min), are also described. Capacity measures that are used in theselection of the location of constellation points include, but are notlimited to, parallel decode (PD) capacity and joint capacity.

In many embodiments, the communication systems utilize capacityapproaching codes including, but not limited to, LDPC and Turbo codes.As is discussed further below, direct optimization of the constellationpoints of a communication system utilizing a capacity approachingchannel code, can yield different constellations depending on the SNRfor which they are optimized. Therefore, the same constellation isunlikely to achieve the same gains applied across all code rates; thatis, the same constellation will not enable the best possible performanceacross all rates. In many instances, a constellation at one code ratecan achieve gains that cannot be achieved at another code rate.Processes for selecting capacity optimized constellations to achieveincreased gains based upon a specific coding rate in accordance withembodiments of the invention are described below. Constellations pointsfor geometric PAM-8, PAM-16, and PAM-32 constellations that areoptimized for joint capacity or PD capacity at specific SNRs are alsoprovided. Additional geometric PAM-8, PAM-16, and PAM-32 constellationsthat are probabilistically likely to provide performance gains comparedto constellations that maximize d_(min), which were identified byperturbing the constellation points of geometric PAM-8, PAM-16, andPAM-32 constellations optimized for joint capacity or PD capacity, arealso described. The constellations are described as beingprobabilistically likely to provide performance gains, because allpossible constellations within the ranges have not been exhaustivelysearched. Within each disclosed range, a large number of constellationswere selected at random, and it was verified that all the selectedconstellations provided a gain that exceeds the given percentage of theoptimal capacity increase achieved by the optimized constellationsrelative to a constellation that maximizes d_(min) (i.e. a PAM equallyspaced constellation). In this way, ranges that are probabilisticallylikely to provide a performance gain that is at least a predeterminedpercentage of the optimal increase in capacity can be identified and aspecific geometric constellation can be compared against the ranges as aguide to the increase in capacity that is likely to be achieved. In anumber of embodiments, the communication systems can adapt the locationof points in a constellation in response to channel conditions, changesin code rate and/or to change the target user data rate.

Communication Systems

A communication system in accordance with an embodiment of the inventionis shown in FIG. 1. The communication system 10 includes a source 12that provides user bits to a transmitter 14. The transmitter transmitssymbols over a channel to a receiver 16 using a predetermined modulationscheme. The receiver uses knowledge of the modulation scheme, to decodethe signal received from the transmitter. The decoded bits are providedto a sink device that is connected to the receiver.

A transmitter in accordance with an embodiment of the invention is shownin FIG. 2. The transmitter 14 includes a coder 20 that receives userbits from a source and encodes the bits in accordance with apredetermined coding scheme. In a number of embodiments, a capacityapproaching code such as a turbo code or a LDPC code is used. In otherembodiments, other coding schemes can be used to providing a coding gainwithin the communication system. A mapper 22 is connected to the coder.The mapper maps the bits output by the coder to a symbol within ageometrically distributed signal constellation stored within the mapper.The mapper provides the symbols to a modulator 24, which modulates thesymbols for transmission via the channel.

A receiver in accordance with an embodiment of the invention isillustrated in FIG. 3. The receiver 16 includes a demodulator 30 thatdemodulates a signal received via the channel to obtain symbol or bitlikelihoods. The demapper uses knowledge of the geometrically shapedsymbol constellation used by the transmitter to determine theselikelihoods. The demapper 32 provides the likelihoods to a decoder 34that decodes the encoded bit stream to provide a sequence of receivedbits to a sink.

Geometrically Shaped Constellations

Transmitters and receivers in accordance with embodiments of theinvention utilize geometrically shaped symbol constellations. In severalembodiments, a geometrically shaped symbol constellation is used thatoptimizes the capacity of the constellation. In many embodiments,geometrically shaped symbol constellations, which include constellationpoints within predetermined ranges of the constellation points of acapacity optimized constellation, and that provide improved capacitycompared to constellations that maximize d_(min) are used. Variousgeometrically shaped symbol constellations that can be used inaccordance with embodiments of the invention, techniques for derivinggeometrically shaped symbol constellations are described below.

Selection of a Geometrically Shaped Constellation

Selection of a geometrically shaped constellation for use in acommunication system in accordance with an embodiment of the inventioncan depend upon a variety of factors including whether the code rate isfixed. In many embodiments, a geometrically shaped constellation is usedto replace a conventional constellation (i.e. a constellation maximizedfor d_(min)) in the mapper of transmitters and the demapper of receiverswithin a communication system. Upgrading a communication system involvesselection of a constellation and in many instances the upgrade can beachieved via a simple firmware upgrade. In other embodiments, ageometrically shaped constellation is selected in conjunction with acode rate to meet specific performance requirements, which can forexample include such factors as a specified bit rate, a maximum transmitpower. Processes for selecting a geometric constellation when upgradingexisting communication systems and when designing new communicationsystems are discussed further below.

Upgrading Existing Communication Systems

A geometrically shaped constellation that provides a capacity, which isgreater than the capacity of a constellation maximized for d_(min), canbe used in place of a conventional constellation in a communicationsystem in accordance with embodiments of the invention. In manyinstances, the substitution of the geometrically shaped constellationcan be achieved by a firmware or software upgrade of the transmittersand receivers within the communication system. Not all geometricallyshaped constellations have greater capacity than that of a constellationmaximized for d_(min). One approach to selecting a geometrically shapedconstellation having a greater capacity than that of a constellationmaximized for d_(min) is to optimize the shape of the constellation withrespect to a measure of the capacity of the constellation for a givenSNR. Another approach is to select a constellation from a range that isprobabilistically likely to yield a constellation having at least apredetermined percentage of the optimal capacity increase. Such anapproach can prove useful in circumstances, for example, where anoptimized constellation is unable to be implemented. Capacity measuresthat can be used in the optimization process can include, but are notlimited to, joint capacity or parallel decoding capacity.

Joint Capacity and Parallel Decoding Capacity

A constellation can be parameterized by the total number ofconstellation points, M, and the number of real dimensions, d_(min). Insystems where there are no belief propagation iterations between thedecoder and the constellation bit demapper, the constellation demappercan be thought of as part of the channel. A diagram conceptuallyillustrating the portions of a communication system that can beconsidered part of the channel for the purpose of determining PDcapacity is shown in FIG. 4a . The portions of the communication systemthat are considered part of the channel are indicated by the ghost line40. The capacity of the channel defined as such is the parallel decoding(PD) capacity, given by:

$C_{PD} = {\sum\limits_{i = 0}^{l - 1}\;{I\left( {X_{i};Y} \right)}}$where X_(i) is the ith bit of the l-bits transmitted symbol, and Y isthe received symbol, and I(A;B) denotes the mutual information betweenrandom variables A and B.

Expressed another way, the PD capacity of a channel can be viewed interms of the mutual information between the output bits of the encoder(such as an LDPC encoder) at the transmitter and the likelihoodscomputed by the demapper at the receiver. The PD capacity is influencedby both the placement of points within the constellation and by thelabeling assignments.

With belief propagation iterations between the demapper and the decoder,the demapper can no longer be viewed as part of the channel, and thejoint capacity of the constellation becomes the tightest known bound onthe system performance. A diagram conceptually illustrating the portionsof a communication system that are considered part of the channel forthe purpose of determining the joint capacity of a constellation isshown in FIG. 4b . The portions of the communication system that areconsidered part of the channel are indicated by the ghost line 42. Thejoint capacity of the channel is given by:C _(JOINT) =I(X;Y)

Joint capacity is a description of the achievable capacity between theinput of the mapper on the transmit side of the link and the output ofthe channel (including for example AWGN and Fading channels). Practicalsystems must often ‘demap’ channel observations prior to decoding. Ingeneral, the step causes some loss of capacity. In fact it can be proventhat C_(G)≧C_(JOINT)≧C_(PD). That is, C_(JOINT) upper bounds thecapacity achievable by C_(PD). The methods of the present invention aremotivated by considering the fact that practical limits to a givencommunication system capacity are limited by C_(JOINT) and C_(PD). Inseveral embodiments of the invention, geometrically shapedconstellations are selected that maximize these measures.

Selecting a Constellation Having an Optimal Capacity

Geometrically shaped constellations in accordance with embodiments ofthe invention can be designed to optimize capacity measures including,but not limited to PD capacity or joint capacity. A process forselecting the points, and potentially the labeling, of a geometricallyshaped constellation for use in a communication system having a fixedcode rate in accordance with an embodiment of the invention is shown inFIG. 5. The process 50 commences with the selection (52) of anappropriate constellation size M and a desired capacity per dimension η.In the illustrated embodiment, the process involves a check (52) toensure that the constellation size can support the desired capacity. Inthe event that the constellation size could support the desiredcapacity, then the process iteratively optimizes the M-ary constellationfor the specified capacity. Optimizing a constellation for a specifiedcapacity often involves an iterative process, because the optimalconstellation depends upon the SNR at which the communication systemoperates. The SNR for the optimal constellation to give a requiredcapacity is not known a priori. Throughout the description of thepresent invention SNR is defined as the ratio of the averageconstellation energy per dimension to the average noise energy perdimension. In most cases the capacity can be set to equal the targetuser bit rate per symbol per dimension. In some cases adding someimplementation margin on top of the target user bit rate could result ina practical system that can provide the required user rate at a lowerSNR. The margin is code dependent. The following procedure could be usedto determine the target capacity that includes some margin on top of theuser rate. First, the code (e.g. LDPC or Turbo) can be simulated inconjunction with a conventional equally spaced constellation. Second,from the simulation results the actual SNR of operation at the requirederror rate can be found. Third, the capacity of the conventionalconstellation at that SNR can be computed. Finally, a geometricallyshaped constellation can be optimized for that capacity.

In the illustrated embodiment, the iterative optimization loop involvesselecting an initial estimate of the SNR at which the system is likelyto operate (i.e. SNR_(in)). In several embodiments the initial estimateis the SNR required using a conventional constellation. In otherembodiments, other techniques can be used for selecting the initial SNR.An M-ary constellation is then obtained by optimizing (56) theconstellation to maximize a selected capacity measure at the initialSNR_(in) estimate. Various techniques for obtaining an optimizedconstellation for a given SNR estimate are discussed below.

The SNR at which the optimized M-ary constellation provides the desiredcapacity per dimension □ (SNR_(out)) is determined (57). A determination(58) is made as to whether the SNR_(out) and SNR_(in) have converged. Inthe illustrated embodiment convergence is indicated by SNR_(out)equaling SNR_(in). In a number of embodiments, convergence can bedetermined based upon the difference between SNR_(out) and SNR_(in)being less than a predetermined threshold. When SNR_(out) and SNR_(in)have not converged, the process performs another iteration selectingSNR_(out) as the new SNR_(in) (55). When SNR_(out) and SNR_(in) haveconverged, the capacity measure of the constellation has been optimized.As is explained in more detail below, capacity optimized constellationsat low SNRs are geometrically shaped constellations that can achievesignificantly higher performance gains (measured as reduction in minimumrequired SNR) than constellations that maximize d_(min).

The process illustrated in FIG. 5 can maximize PD capacity or jointcapacity of an M-ary constellation for a given SNR. Although the processillustrated in FIG. 5 shows selecting an M-ary constellation optimizedfor capacity, a similar process could be used that terminates upongeneration of an M-ary constellation where the SNR gap to Gaussiancapacity at a given capacity is a predetermined margin lower than theSNR gap of a conventional constellation, for example 0.5 db.Alternatively, other processes that identify M-ary constellations havingcapacity greater than the capacity of a conventional constellation canbe used in accordance with embodiments of the invention. For example,the effect of perturbations on the constellation points of optimizedconstellations can be used to identify ranges in which predeterminedperformance improvements are probabilistically likely to be obtained.The ranges can then be used to select geometrically shapedconstellations for use in a communication system. A geometrically shapedconstellation in accordance with embodiments of the invention canachieve greater capacity than the capacity of a constellation thatmaximizes d_(min) without having the optimal capacity for the SNR rangewithin which the communication system operates.

We note that constellations designed to maximize joint capacity may alsobe particularly well suited to codes with symbols over GF(q), or withmulti-stage decoding. Conversely constellations optimized for PDcapacity could be better suited to the more common case of codes withsymbols over GF(2)

Optimizing the Capacity of an M-Ary Constellation at a Given SNR

Processes for obtaining a capacity optimized constellation often involvedetermining the optimum location for the points of an M-aryconstellation at a given SNR. An optimization process, such as theoptimization process 56 shown in FIG. 5, typically involvesunconstrained or constrained non-linear optimization. Possible objectivefunctions to be maximized are the Joint or PD capacity functions. Thesefunctions may be targeted to channels including but not limited toAdditive White Gaussian Noise (AWGN) or Rayleigh fading channels. Theoptimization process gives the location of each constellation pointidentified by its symbol labeling. In the case where the objective isjoint capacity, point bit labelings are irrelevant meaning that changingthe bit labelings doesn't change the joint capacity as long as the setof point locations remains unchanged.

The optimization process typically finds the constellation that givesthe largest PD capacity or joint capacity at a given SNR. Theoptimization process itself often involves an iterative numericalprocess that among other things considers several constellations andselects the constellation that gives the highest capacity at a givenSNR. In other embodiments, the constellation that requires the least SNRto give a required PD capacity or joint capacity can also be found. Thisrequires running the optimization process iteratively as shown in FIG.5.

Optimization constraints on the constellation point locations mayinclude, but are not limited to, lower and upper bounds on pointlocation, peak to average power of the resulting constellation, and zeromean in the resulting constellation. It can be easily shown that aglobally optimal constellation will have zero mean (no DC component).Explicit inclusion of a zero mean constraint helps the optimizationroutine to converge more rapidly. Except for cases where exhaustivesearch of all combinations of point locations and labelings is possibleit will not necessarily always be the case that solutions are provablyglobally optimal. In cases where exhaustive search is possible, thesolution provided by the non-linear optimizer is in fact globallyoptimal.

The processes described above provide examples of the manner in which ageometrically shaped constellation having an increased capacity relativeto a conventional capacity can be obtained for use in a communicationsystem having a fixed code rate and modulation scheme. The actual gainsachievable using constellations that are optimized for capacity comparedto conventional constellations that maximize d_(min) are consideredbelow.

Gains Achieved by Optimized Geometrically Spaced Constellations

The ultimate theoretical capacity achievable by any communication methodis thought to be the Gaussian capacity, C_(G) which is defined as:C _(G)=½ log₂(1+SNR)

Where signal-to-noise (SNR) is the ratio of expected signal power toexpected noise power. The gap that remains between the capacity of aconstellation and C_(G) can be considered a measure of the quality of agiven constellation design.

The gap in capacity between a conventional modulation scheme incombination with a theoretically optimal coder can be observed withreference to FIGS. 6a and 6b . FIG. 6a includes a chart 60 showing acomparison between Gaussian capacity and the PD capacity of conventionalPAM-2, 4, 8, 16, and 32 constellations that maximize d_(min). Gaps 62exist between the plot of Gaussian capacity and the PD capacity of thevarious PAM constellations. FIG. 6b includes a chart 64 showing acomparison between Gaussian capacity and the joint capacity ofconventional PAM-2, 4, 8, 16, and 32 constellations that maximized_(min), Gaps 66 exist between the plot of Gaussian capacity and thejoint capacity of the various PAM constellations. These gaps in capacityrepresent the extent to which conventional PAM constellations fall shortof obtaining the ultimate theoretical capacity i.e. the Gaussiancapacity.

In order to gain a better view of the differences between the curvesshown in FIGS. 6a and 6b at points close to the Gaussian capacity, theSNR gap to Gaussian capacity for different values of capacity for eachconstellation are plotted in FIG. 7. It is interesting to note from thechart 70 in FIG. 7 that (unlike the joint capacity) at the same SNR, thePD capacity does not necessarily increase with the number ofconstellation points. As is discussed further below, this is not thecase with PAM constellations optimized for PD capacity.

FIGS. 8a and 8b summarize performance of constellations for PAM-4, 8,16, and 32 optimized for PD capacity and joint capacity (it should benoted that BPSK is the optimal PAM-2 constellation at all code rates).The constellations are optimized for PD capacity and joint capacity fordifferent target user bits per dimension (i.e. code rates). Theoptimized constellations are different depending on the target user bitsper dimension, and also depending on whether they have been designed tomaximize the PD capacity or the joint capacity. All the PD optimized PAMconstellations are labeled using a gray labeling which is not always thebinary reflective gray labeling. It should be noted that not all graylabels achieve the maximum possible PD capacity even given the freedomto place the constellation points anywhere on the real line. FIG. 8ashows the SNR gap for each constellation optimized for PD capacity. FIG.8b shows the SNR gap to Gaussian capacity for each constellationoptimized for joint capacity. Again, it should be emphasized that each‘+’ on the plot represents a different constellation.

Referring to FIG. 8a , the coding gain achieved using a constellationoptimized for PD capacity can be appreciated by comparing the SNR gap ata user bit rate per dimension of 2.5 bits for PAM-32. A user bit rateper dimension of 2.5 bits for a system transmitting 5 bits per symbolconstitutes a code rate of 1/2. At that code rate the constellationoptimized for PD capacity provides an additional coding gain ofapproximately 1.5 dB when compared to the conventional PAM-32constellation.

The SNR gains that can be achieved using constellations that areoptimized for PD capacity can be verified through simulation. Theresults of a simulation conducted using a rate 1/2 LDPC code inconjunction with a conventional PAM-32 constellation and in conjunctionwith a PAM-32 constellation optimized for PD capacity are illustrated inFIG. 9. A chart 90 includes plots of Frame Error Rate performance of thedifferent constellations with respect to SNR and using different lengthcodes (i.e. k=4,096 and k=16,384). Irrespective of the code that isused, the constellation optimized for PD capacity achieves a gain ofapproximately 1.3 dB, which closely approaches the gain predicted fromFIG. 8 a.

Capacity Optimized PAM Constellations

Using the processes outlined above, locus plots of PAM constellationsoptimized for capacity can be generated that show the location of pointswithin PAM constellations versus SNR. Locus plots of PAM-4, 8, 16, and32 constellations optimized for PD capacity and joint capacity andcorresponding design tables at various typical user bit rates perdimension are illustrated in FIGS. 10a-17b . The locus plots and designtables show PAM-4, 8, 16; and 32 constellation point locations andlabelings from low to high SNR corresponding to a range of low to highspectral efficiency.

In FIG. 10a , a locus plot 100 shows the location of the points of PAM-4constellations optimized for joint capacity plotted against achievedcapacity. A similar locus plot 105 showing the location of the points ofjoint capacity optimized PAM-4 constellations plotted against SNR isincluded in FIG. 10b . In FIG. 10c , the location of points for PAM-4optimized for PD capacity is plotted against achievable capacity and inFIG. 10d the location of points for PAM-4 for PD capacity is plottedagainst SNR. At low SNRs, the PD capacity optimized PAM-4 constellationshave only 2 unique points, while the joint capacity optimizedconstellations have 3. As SNR is increased, each optimization eventuallyprovides 4 unique points. This phenomenon is explicitly described inFIG. 11a and FIG. 11b where vertical slices of FIGS. 10 ab and 10 cd arecaptured in tables describing some PAM-4 constellations designs ofinterest. The SNR slices selected represent designs that achievecapacities={0.5, 0.75, 1.0, 1.25, 1.5} bits per symbol (bps). Given thatPAM-4 can provide at most log₂(4)=2 bps, these design points representsystems with information code rates R={1/4, 3/8, 1/2, 5/8, 3/4}respectively.

FIGS. 12 ab and 12 cd present locus plots of PD capacity and jointcapacity optimized PAM-8 constellation points versus achievable capacityand SNR. FIGS. 13a and 13b provide slices from these plots at SNRscorresponding to achievable capacities η={0.5, 1.0, 1.5, 2.0, 2.5} bps.Each of these slices correspond to systems with code rate R=ηbps/log₂(8), resulting in R={1/6, 1/3, 1/2, 2/3, 5/6}. As an example ofthe relative performance of the constellations in these tables, considerFIG. 13b which shows a PD capacity optimized PAM-8 constellationoptimized for SNR=9.00 dB, or 1.5 bps. We next examine the plot providedin FIG. 8a and see that the gap of the optimized constellation to theultimate, Gaussian, capacity (CG) is approximately 0.5 dB. At the samespectral efficiency, the gap of the traditional PAM-8 constellation isapproximately 1.0 dB. The advantage of the optimized constellation is0.5 dB for the same rate (in this case R=1/2). This gain can be obtainedby only changing the mapper and demapper in the communication system andleaving all other blocks the same.

Similar information is presented in FIGS. 14a-14d, and 15a-15b whichprovide loci plots and design tables for PAM-16 PD capacity and jointcapacity optimized constellations. Likewise FIGS. 16a-16d, 17a and 17bprovide loci plots and design tables for PAM-32 PD capacity and jointcapacity optimized constellations.

Capacity Optimized PSK Constellations Over the AWGN Channel

Traditional phase shift keyed (PSK) constellations are already quiteoptimal. This can be seen in the chart 180 comparing the SNR gaps oftradition PSK with capacity optimized PSK constellations shown in FIG.18 where the gap between PD capacity and Gaussian capacity is plottedfor traditional PSK-4, 8, 16, and 32 and for PD capacity optimizedPSK-4, 8, 16, and 32.

The locus plot of PD optimized PSK-32 points across SNR is shown in FIG.19, which actually characterizes all PSKs with spectral efficiency η≦5.This can be seen in FIG. 20. Note that at low SNR (0.4 dB) the optimalPSK-32 design is the same as traditional PSK-4, at SNR=8.4 dB optimalPSK-32 is the same as traditional PSK-8, at SNR=14.8 dB optimal PSK-32is the same as traditional PSK-16, and finally at SNRs greater than 20.4dB optimized PSK-32 is the same as traditional PSK-32. There are SNRsbetween these discrete points (for instance SNR=2 and 15 dB) for whichoptimized PSK-32 provides superior PD capacity when compared totraditional PSK constellations.

We note now that the locus of points for PD optimized PSK-32 in FIG. 19in conjunction with the gap to Gaussian capacity curve for optimizedPSK-32 in FIG. 18 implies a potential design methodology. Specifically,the designer could achieve performance equivalent or better than thatenabled by traditional PSK-4,8, and 16 by using only the optimizedPSK-32 in conjunction with a single tuning parameter that controlledwhere the constellation points should be selected from on the locus ofFIG. 19. Such an approach would couple a highly rate adaptive channelcode that could vary its rate, for instance, rate 4/5 to achieve andoverall (code plus optimized PSK-32 modulation) spectral efficiency of 4bits per symbol, down to 1/5 to achieve an overall spectral efficiencyof 1 bit per symbol. Such an adaptive modulation and coding system couldessentially perform on the optimal continuum represented by therightmost contour of FIG. 18.

Adaptive Rate Design

In the previous example spectrally adaptive use of PSK-32 was described.Techniques similar to this can be applied for other capacity optimizedconstellations across the link between a transmitter and receiver. Forinstance, in the case where a system implements quality of service it ispossible to instruct a transmitter to increase or decrease spectralefficiency on demand. In the context of the current invention a capacityoptimized constellation designed precisely for the target spectralefficiency can be loaded into the transmit mapper in conjunction with acode rate selection that meets the end user rate goal. When such amodulation/code rate change occurred a message could propagated to thereceiver so that the receiver, in anticipation of the change, couldselect a demapper/decoder configuration in order to match the newtransmit-side configuration.

Conversely, the receiver could implement a quality of performance basedoptimized constellation/code rate pair control mechanism. Such anapproach would include some form of receiver quality measure. This couldbe the receiver's estimate of SNR or bit error rate. Take for examplethe case where bit error rate was above some acceptable threshold. Inthis case, via a backchannel, the receiver could request that thetransmitter lower the spectral efficiency of the link by swapping to analternate capacity optimized constellation/code rate pair in the coderand mapper modules and then signaling the receiver to swap in thecomplementary pairing in the demapper/decoder modules.

Geometrically Shaped QAM Constellations

Quadrature amplitude modulation (QAM) constellations can be constructedby orthogonalizing PAM constellations into QAM in phase and quadraturecomponents. Constellations constructed in this way can be attractive inmany applications because they have low-complexity demappers.

In FIG. 21 we provide an example of a Quadrature Amplitude Modulationconstellation constructed from a Pulse Amplitude Modulationconstellation. The illustrated embodiment was constructed using a PAM-8constellation optimized for PD capacity for the AWGN channel at user bitrate per dimension of 1.5 bits (corresponds to an SNR of 9.0 dB) (seeFIG. 13b ). The label-point pairs in this PAM-8 constellation are {(000,−1.72), (001, −0.81), (010, 1.72), (011,−0.62), (100, 0.62), (101,0.02), (110, 0.81), (111, −0.02)}. Examination of FIG. 21 shows that theQAM constellation construction is achieved by replicating a complete setof PAM-8 points in the quadrature dimension for each of the 8 PAM-8points in the in-phase dimension. Labeling is achieved by assigning thePAM-8 labels to the LSB range on the in-phase dimension and to the MSBrange on the quadrature dimension. The resulting 8×8 outer product formsa highly structured QAM-64 for which very low-complexity de-mappers canbe constructed. Due to the orthogonality of the in-phase and quadraturecomponents the capacity characteristics of the resulting QAM-64constellation are identical to that of the PAM-8 constellation on aper-dimension basis. The same process can be applied for constellationsoptimized for the Raleigh fading channel.

N-Dimensional Constellation Optimization

Rather than designing constellations in 1-D (PAM for instance) and thenextending to 2-D (QAM), it is possible to take direct advantage in theoptimization step of the additional degree of freedom presented by anextra spatial dimension. In general it is possible to designN-dimensional constellations and associated labelings. The complexity ofthe optimization step grows exponentially in the number of dimensions asdoes the complexity of the resulting receiver de-mapper. Suchconstructions constitute embodiments of the invention and simply requiremore ‘run-time’ to produce.

Capacity Optimized Constellations for Fading Channels

Similar processes to those outlined above can be used to design capacityoptimized constellations for fading channels in accordance withembodiments of the invention. The processes are essentially the samewith the exception that the manner in which capacity is calculated ismodified to account for the fading channel. A fading channel can bedescribed using the following equation:Y=a(t)·X+Nwhere X is the transmitted signal, N is an additive white Gaussian noisesignal and a(t) is the fading distribution, which is a function of time.

In, the case of a fading channel, the instantaneous SNR at the receiverchanges according to a fading distribution. The fading distribution isRayleigh and has the property that the average SNR of the system remainsthe same as in the case of the AWGN channel, E[X²]/E[N²]. Therefore, thecapacity of the fading channel can be computed by taking the expectationof AWGN capacity, at a given average SNR, over a fading distribution ofa, such as the Raleigh fading distribution, that drives the distributionof the instantaneous SNR.

Many fading channels follow a Rayleigh distribution. FIGS. 22a-24b arelocus plots of PAM-4, 8, and 16 constellations that have been optimizedfor PD capacity on a Rayleigh fading channel. Locus plots versus userbit rate per dimension and versus SNR are provided. Similar processescan be used to obtain capacity optimized constellations that areoptimized using other capacity measures, such as joint capacity, and/orusing different modulation schemes.

Geometric PAM-8, PAM-16, and PAM-32 Constellations

As described above, geometric constellations can be obtained that areoptimized for joint or PD capacity at specific SNRs. In addition, rangescan be specified for the constellation points of a geometricconstellation that are probabilistically likely to result in geometricconstellations that provide at least a predetermined performanceimprovement relative to a constellation that maximizes d_(min). Turningnow to FIGS. 25-95, geometric PAM-8, PAM-16, and PAM-32 constellationsoptimized for joint and PD capacity over the Raleigh fading channel atspecific SNRs are listed. The performances of the optimal constellationsare compared to the performances of traditional constellations thatmaximize d_(min). Ranges for the constellation points are also definedat specific SNRs, where constellations having constellation pointsselected from within the ranges are probabilistically likely (withprobability close to one) to result in at least a predeterminedperformance improvement at the specified SNR relative to a traditionalconstellation that maximizes d_(min).

The geometric constellations disclosed in FIGS. 25-95 are defined bypoints y(i) such that y(i)=k(x(i)+r(i))+c. Values for x(i) and bounds onr(i) are provided in FIGS. 25-95 for PAM-8, PAM-16, and PAM-32 optimizedfor joint and PD capacity under Rayleigh fading channel conditions. ForPAM-8 0≦i≦7, PAM-16 0≦i≦15, and for PAM-32 0≦i≦31. To achieve optimalpower efficiency, c should be set to zero. The constant k can be viewedas a scaling factor that only changes the power of the constellation.For example, the constellations disclosed in FIGS. 25-95 were scaled toprovide constellations with an arbitrary power of 10,912. The sameconstellations can be represented using any other arbitrary scalingincluding (but not limited) to scaling the disclosed constellations tohave an arbitrary power of 16. To scale the provided constellations withan arbitrary power of 10,912 to have an arbitrary power of 16, a scalingfactor of 1/26.1151297144012 can be used. By way of example,constellation design 1 from the tables in FIGS. 86-93 can be scaled to apower of 16 as follows:

Constellation Design 1 Power = 10,912 Power = 16 −37.9896 −1.4547−30.4644 −1.1665 −25.8467 −0.9897 −26.1314 −1.0006 −17.8829 −0.6848−18.1165 −0.6937 −18.8919 −0.7234 −18.8919 −0.7234 −4.8867 −0.1871−4.9004 −0.1876 −4.9384 −0.1891 −4.9384 −0.1891 −8.2966 −0.3177 −8.2966−0.3177 −8.2966 −0.3177 −8.2966 −0.3177 37.9896 1.4547 30.4644 1.166525.8467 0.9897 26.1314 1.0006 17.884 0.6848 18.1156 0.6937 18.89180.7234 18.8918 0.7234 4.8863 0.1871 4.9008 0.1877 4.9384 0.1891 4.93840.1891 8.2966 0.3177 8.2966 0.3177 8.2966 0.3177 8.2966 0.3177Similarly, modifying k also scales the resulting maximum ranges forconstellation design 1 as shown in the tables from FIGS. 94-95 asfollows:

Constellation Design 1 Percentage of Optimal Capacity Gain 5% 20% 50%90% 100% Maximum ranges ±2.1890 ±1.9700 ±1.5960 ±0.6870 ±0 where k = 1Maximum ranges ±8.3821 × 10⁻² ±7.5435 × 10⁻² ±6.1114 × 10⁻² ±2.6307 ×10⁻² ±0 where k = 1/26.1151297144012By way of further example, constellation design 8 from the tables inFIGS. 86-93 can be scaled to a power of 16 as follows:

Constellation Design 8 Power = 10,912 Power = 16 −38.8088 −1.4861−31.7370 −1.2153 −25.0630 −0.9597 −26.1892 −1.0028 −16.6675 −0.6382−16.6894 −0.6391 −18.4668 −0.7071 −18.4668 −0.7071 −3.7549 −0.1438−3.7549 −0.1438 −3.7549 −0.1438 −3.7549 −0.1438 −9.2115 −0.3527 −9.2115−0.3527 −9.0480 −0.3465 −9.0888 −0.3480 38.8190 1.4865 31.7383 1.215325.0617 0.9597 26.1893 1.0028 16.6677 0.6382 16.6875 0.6390 18.46620.7071 18.4662 0.7071 3.7539 0.1437 3.7539 0.1437 3.7539 0.1437 3.75390.1437 9.2107 0.3527 9.2107 0.3527 9.0476 0.3465 9.0509 0.3466Similarly, modifying k also scales the resulting maximum ranges forconstellation design 8 as shown in the tables from FIGS. 94-95 asfollows:

Constellation Design 8 Percentage of Optimal Capacity Gain 5% 20% 50%90% 100% Maximum ranges ±1.9640 ±1.7680 ±1.2890 ±0.6020 ±0 where k = 1Maximum ranges ±7.5205 × 10⁻² ±6.7700 ± 10⁻² ±4.9358 × 10⁻² ±2.3052 ±10⁻² ±0 where k = 1/26.1151297144012By way of further example, constellation design 13 from the tables inFIGS. 86-93 can be scaled to a power of 16 as follows:

Constellation Design 13 Power = 10,912 Power = 16 −39.2317 −1.5023−32.0637 −1.2278 −24.6366 −0.9434 −26.214 −1.0038 −15.9554 −0.6110−15.9554 −0.6110 −18.7399 −0.7176 −18.5633 −0.7108 −3.3634 −0.1288−3.3645 −0.1288 −3.3963 −0.1301 −3.3963 −0.1301 −9.556 −0.3659 −9.556−0.3659 −8.9797 −0.3439 −8.9852 −0.3441 39.2363 1.5024 32.0635 1.227824.6352 0.9433 26.2126 1.0037 15.9543 0.6109 15.9543 0.6109 18.73810.7175 18.5618 0.7108 3.3646 0.1288 3.3653 0.1289 3.3966 0.1301 3.39660.1301 9.5562 0.3659 9.5562 0.3659 8.9803 0.3439 8.9851 0.3441Similarly, modifying k also scales the resulting maximum ranges forconstellation design 13 as follows:

Constellation Design 13 Percentage of Optimal Capacity Gain 5% 20% 50%90% 100% Maximum ranges ±1.7080 ±1.5370 ±1.2450 ±0.5190 ±0 where k = 1Maximum ranges ±6.5403 × 10⁻² ±5.8855 × 10⁻² ±4.7674 × 10⁻² ±1.9874 ×10⁻² ±0 where k = 1/26.1151297144012By way of further example, constellation design 19 from the tables inFIGS. 86-93 can be scaled to a power of 16 as follows:

Constellation Design 19 Power = 10,912 Power = 16 −39.1811 −1.5003−32.19 −1.2326 −24.2565 −0.9288 −26.6863 −1.0219 −15.3045 −0.5860−15.3045 −0.5860 −19.4031 −0.7430 −18.8423 −0.7215 −2.6518 −0.1015−2.652 −0.1016 −3.5662 −0.1366 −3.5662 −0.1366 −10.1865 −0.3901 −10.1865−0.3901 −8.2708 −0.3167 −8.2773 −0.3170 39.1856 1.5005 32.1917 1.232724.2575 0.9289 26.686 1.0219 15.3039 0.5860 15.3039 0.5860 19.40190.7429 18.8439 0.7216 2.651 0.1015 2.6514 0.1015 3.5662 0.1366 3.56620.1366 10.186 0.3900 10.186 0.3900 8.2697 0.3167 8.2746 0.3169Similarly, modifying k also scales the resulting maximum ranges forconstellation design 19 as shown in the tables from FIGS. 94-95 asfollows:

Constellation Design 19 Percentage of Optimal Capacity Gain 5% 20% 50%90% 100% Maximum ranges ±1.3880 ±1.2490 ±1.0120 ±0.4430 ±0 where k = 1Maximum ranges ±5.3145 × 10⁻² ±4.7827 × 10⁻² ±3.8751 × 10⁻² ±1.6963 ×10⁻² ±0 where k = 1/26.1151297144012By way of further example, constellation design 24 from the tables inFIGS. 86-93 can be scaled to a power of 16 as follows:

Constellation Design 24 Power = 10,912 Power = 16 −38.5955 −1.4779−32.0616 −1.2277 −24.0453 −0.9207 −27.0615 −1.0362 −15.1509 −0.5802−15.3494 −0.5878 −19.9129 −0.7625 −18.8201 −0.7207 −1.9769 −0.0757−1.9778 −0.0757 −4.2608 −0.1632 −4.2608 −0.1632 −10.928 −0.4185 −10.9088−0.4177 −8.0512 −0.3083 −8.0512 −0.3083 38.5954 1.4779 32.0594 1.227624.0456 0.9208 27.0586 1.0361 15.1515 0.5802 15.3492 0.5878 19.91270.7625 18.8213 0.7207 1.9789 0.0758 1.9801 0.0758 4.2617 0.1632 4.26170.1632 10.9264 0.4184 10.9084 0.4177 8.0509 0.3083 8.0509 0.3083Similarly, modifying k also scales the resulting maximum ranges forconstellation design 24 as shown in the tables from FIGS. 94-95 asfollows:

Constellation Design 24 Percentage of Optimal Capacity Gain 5% 20% 50%90% 100% Maximum ranges ±1.0640 ±0.9580 ±0.7760 ±0.3410 ±0 where k = 1Maximum ranges ±4.0743 × 10⁻² ±3.6684 × 10⁻² ±2.9715 × 10⁻² ±1.3058 ×10⁻² ±0 where k = 1/26.1151297144012By way of further example, constellation design 26 from the tables inFIGS. 86-93 can be scaled to a power of 16 as follows:

Constellation Design 26 Power = 10,912 Power = 16 −38.3887 −1.4700−32.0467 −1.2271 −24.023 −0.9199 −27.2059 −1.0418 −15.0232 −0.5753−15.4053 −0.5899 −20.1597 −0.7720 −18.7135 −0.7166 −1.854 −0.0710−1.8543 −0.0710 −4.4028 −0.1686 −4.4028 −0.1686 −11.086 −0.4245 −11.0231−0.4221 −8.0244 −0.3073 −8.0271 −0.3074 38.3284 1.4677 32.0412 1.226924.0205 0.9198 27.2021 1.0416 15.026 0.5754 15.4055 0.5899 20.15780.7719 18.7141 0.7166 1.8561 0.0711 1.8565 0.0711 4.4049 0.1687 4.40490.1687 11.0873 0.4246 11.0254 0.4222 8.0268 0.3074 8.0295 0.3075Similarly, modifying k also scales the resulting maximum ranges forconstellation design 26 as shown in the tables from FIGS. 94-95 asfollows:

Constellation Design 26 Percentage of Optimal Capacity Gain 5% 20% 50%90% 100% Maximum ranges ±1.0380 ±0.9340 ±0.7560 ±0.3100 ±0 where k = 1Maximum ranges ±3.9747 × 10⁻² ±3.5765 × 10⁻² ±2.8949 × 10⁻² ±1.1871 ×10⁻² ±0 where k = 1/26.1151297144012By way of further example, constellation design 32 from the tables inFIGS. 86-93 can be scaled to a power of 16 as follows:

Constellation Design 32 Power = 10,912 Power = 16 −37.7602 −1.4459−32.0093 −1.2257 −24.2219 −0.9275 −27.5785 −1.0560 −14.4076 −0.5517−15.8834 −0.6082 −20.9075 −0.8006 −18.6078 −0.7125 −1.6412 −0.0628−1.6645 −0.0637 −4.5506 −0.1743 −4.4707 −0.1712 −11.4808 −0.4396−10.7002 −0.4097 −7.6972 −0.2947 −7.9842 −0.3057 37.7537 1.4457 32.00681.2256 24.2204 0.9274 27.577 1.0560 14.4047 0.5516 15.8829 0.6082 20.9060.8005 18.6066 0.7125 1.6435 0.0629 1.6666 0.0638 4.5532 0.1744 4.47310.1713 11.4808 0.4396 10.7014 0.4098 7.7004 0.2949 7.9858 0.3058Similarly, modifying k also scales the resulting maximum ranges forconstellation design 32 as shown in the tables from FIGS. 94-95 asfollows:

Constellation Design 32 Percentage of Optimal Capacity Gain 5% 20% 50%90% 100% Maximum ranges ±0.7300 ±0.6570 ±0.5320 ±0.2540 ±0 where k = 1Maximum ranges ±2.7953 × 10⁻² ±2.5158 × 10⁻² ±2.0371 × 10⁻² ±0.9726 ×10⁻² ±0 where k = 1/26.1151297144012By way of further example, constellation design 37 from the tables inFIGS. 86-93 can be scaled to a power of 16 as follows:

Constellation Design 37 Power = 10,912 Power = 16 −36.8223 −1.4100−31.6346 −1.2114 −24.3912 −0.9340 −27.5952 −1.0567 −14.6073 −0.5593−16.4909 −0.6315 −21.3627 −0.8180 −18.963 −0.7261 −1.1401 −0.0437−2.0669 −0.0791 −5.1427 −0.1969 −4.118 −0.1577 −12.2395 −0.4687 −10.704−0.4099 −7.3701 −0.2822 −8.5986 −0.3293 36.8223 1.4100 31.6352 1.211424.3917 0.9340 27.5958 1.0567 14.6073 0.5593 16.4909 0.6315 21.36310.8180 18.9632 0.7261 1.1397 0.0436 2.0664 0.0791 5.1424 0.1969 4.11770.1577 12.2394 0.4687 10.7038 0.4099 7.3697 0.2822 8.5984 0.3292Similarly, modifying k also scales the resulting maximum ranges forconstellation design 37 as shown in the tables from FIGS. 94-95 asfollows:

Constellation Design 37 Percentage of Optimal Capacity Gain 5% 20% 50%90% 100% Maximum ranges ±0.6050 ±0.5440 ±0.4410 ±0.1930 ±0 where k = 1Maximum ranges ±2.3167 × 10⁻² ±2.0831 × 10⁻² ±1.6887 × 10⁻² ±0.7390 ×10⁻² ±0 where k = 1/26.1151297144012By way of further example, constellation design 43 from the tables inFIGS. 86-93 can be scaled to a power of 16 as follows:

Constellation Design 43 Power = 10,912 Power = 16 −35.7806 −1.3701−31.1542 −1.1930 −24.5029 −0.9383 −27.5041 −1.0532 −15.019 −0.5751−17.0214 −0.6518 −21.7164 −0.8316 −19.3443 −0.7407 −0.9323 −0.0357−2.391 −0.0916 −5.6791 −0.2175 −4.1763 −0.1599 −12.8773 −0.4931 −11.1053−0.4252 −7.5809 −0.2903 −9.1812 −0.3516 35.7805 1.3701 31.1543 1.193024.5029 0.9383 27.5041 1.0532 15.019 0.5751 17.0214 0.6518 21.71640.8316 19.3443 0.7407 0.9322 0.0357 2.391 0.0916 5.6791 0.2175 4.17620.1599 12.8772 0.4931 11.1053 0.4252 7.5809 0.2903 9.1813 0.3516Similarly, modifying k also scales the resulting maximum ranges forconstellation design 43 as shown in the tables from FIGS. 94-95 asfollows:

Constellation Design 43 Percentage of Optimal Capacity Gain 5% 20% 50%90% 100% Maximum ranges ±0.3890 ±0.3500 ±0.2830 ±0.1220 ±0 where k = 1Maximum ranges ±1.4896 × 10⁻² ±1.3402 × 10⁻² ±1.0837 × 10⁻² ±0.4672 ×10⁻² ±0 where k = 1/26.1151297144012By way of further example, constellation design 52 from the tables inFIGS. 86-93 can be scaled to a power of 16 as follows:

Constellation Design 52 Power = 10,912 Power = 16 −34.7205 −1.3295−30.6582 −1.1740 −24.6158 −0.9426 −27.3954 −1.0490 −15.478 −0.5927−17.5199 −0.6709 −22.0414 −0.8440 −19.7371 −0.7558 −0.8935 −0.0342−2.5729 −0.0985 −6.0512 −0.2317 −4.3433 −0.1663 −13.4322 −0.5143−11.5522 −0.4424 −7.8881 −0.3021 −9.6593 −0.3699 34.719 1.3295 30.65731.1739 24.6154 0.9426 27.3948 1.0490 15.4781 0.5927 17.52 0.6709 22.04130.8440 19.7371 0.7558 0.894 0.0342 2.5734 0.0985 6.0516 0.2317 4.34380.1663 13.4325 0.5144 11.5525 0.4424 7.8885 0.3021 9.5697 0.3664Similarly, modifying k also scales the resulting maximum ranges forconstellation design 52 as shown in the tables from FIGS. 94-95 asfollows:

Constellation Design 52 Percentage of Optimal Capacity Gain 5% 20% 50%90% 100% Maximum ranges ±0.2550 ±0.2290 ±0.1850 ±0.0800 ±0 where k = 1Maximum ranges ±0.9764 × 10⁻² ±0.8769 × 10⁻² ±0.7084 × 10⁻² ±0.3063 ×10⁻² ±0 where k = 1/26.1151297144012As one can readily appreciate, modifying k also scales the resultingmaximum ranges for any geometric PAM-32 constellation designs optimizedfor PD Capacity over a Raleigh fading channel at specific SNRs.

In addition to optimized constellations, FIGS. 25-95 specify ranges forthe points of a geometric constellation, where selecting the points of aconstellation from within the ranges is probabilistically likely toprovide a geometric constellation having at least a predeterminedperformance improvement relative to a constellation that maximizesd_(min). The ranges are expressed as a maximum value for theconstellation range parameter, r(i), which specifies the amount by whichthe point x(i) in the constellation is perturbed relative to thelocation of the corresponding point in the optimal constellation. Acommunication system using a constellation formed from constellationspoints selected from within the ranges specified by the maximum value(i.e. −r_(max)≦r(i)≦r_(max)) is probabilistically likely to achieve apredetermined performance improvement relative to a constellation thatmaximizes d_(min). The predetermined performance improvements associatedwith the ranges specified in FIGS. 25-95 are expressed in terms of apercentage of the increase in capacity achieved by the optimizedconstellation relative to a constellation that maximizes d_(min).Constellations formed from constellation points selected from within theranges are probabilistically likely to achieve an increase in capacityat least as great as the indicated percentage.

With regard to the specific tables shown in FIGS. 25-95, each table isone of three different types of table. A first set of tables shows theperformance of specific geometric constellations optimized for jointcapacity or PD capacity. These tables include 6 columns. The firstcolumn enumerates a design number. The second column provides the SNR atwhich the constellation was optimized for the design defined by theentry in the first column. The third column provides the capacityachieved by the optimized constellation (Opt. Cap) at the SNR given inthe second column. The fourth column provides the capacity achieved(Std. Cap) by a traditional uniformly spaced constellation i.e. a PAMconstellation that maximizes d_(min) (with the same number of points asthe optimized constellation and where binary reflective gray labeling isassumed) at the SNR given in the second column. The fifth column showsthe gain in bits per transmission provided by the optimizedconstellation over a constellation that maximizes d_(min). The sixthcolumn shows the percentage gain in capacity provided by the optimizedconstellation over the capacity provided by the traditional uniformlyspaced constellation.

A second set of tables lists the constellation points of the designsindicated in the first set of tables. These tables contain 9 columns.The first column enumerates a design number. The remaining 8 columnsdescribe a constellation point x(i) enumerated by label in the secondrow of the table. Labels are given in decimal number format. With PAM 8as an example, a label of 011 is given as the decimal number 3.

The third set of tables specifies maximum perturbation ranges for thecapacity optimized constellations indicated in the first set of tables,where the maximum ranges correspond to a high probabilistic likelihoodof at least a predetermined performance improvement relative to aconstellation that maximizes d_(min). These tables contain 8 columns.The first enumerates a design number (corresponding to a design from oneof the aforementioned tables). The second column provides the SNR forthe design defined by the entry in the first column. The remaining 5columns describe parameter r_(max) which is the maximum amount any pointin the designed constellation may be perturbed (in either the positiveor negative direction) and still retain, with probability close tounity, at least the gain noted by each column header of the joint or PDcapacity as a percentage of the gain provided by the correspondingoptimized point design over a traditional constellation that maximizesd_(min) (all at the given SNR). Each table has a last column showingthat if 100% of the gain afforded by the optimized constellation isdesired, then parameter r(i) must be equal to zero (no deviation fromdesigned points described in the point specification tables).

Labelling of Constellations Using Cyclically Rotated Binary ReflectiveGray Labels

In performing optimization with respect to PD capacity, a conjecture canbe made that constraining the optimization process to the subsequentlydescribed class of labelings results in no or negligible loss in PDcapacity (the maximum observed loss is 0.005 bits, but in many casesthere is no loss at all). Use of this labeling constraint speeds theoptimization process considerably. We note that joint capacityoptimization is invariant to choice of labeling. Specifically, jointcapacity depends only on point locations whereas PD capacity depends onpoint locations and respective labelings.

The class of cyclically rotated binary reflective gray labels can beused. The following example, using constellations with cardinality 8,describes the class of cyclically rotated binary reflective gray labels.Given for example the standard gray labeling scheme for

PAM-8:

000, 001, 011, 010, 110, 111, 101, 100

Application of a cyclic rotation, one step left, yields:

001, 011, 010, 110, 111, 101, 100, 000

Application of a cyclic rotation, two steps left, yields:

011, 010, 110, 111, 101, 100, 000, 001

For a constellation with cardinality 8, cyclic rotations of 0 to 7 stepscan be applied. It should be noted that within this class of labelings,some labelings perform better than others. It should also be noted thatdifferent rotations may yield labelings that are equivalent (throughtrivial column swapping and negation operations). In general, labelingscan be expressed in different but equivalent forms through trivialoperations such as column swapping and negation operations. For examplethe binary reflective gray labels with one step rotation:

001, 011, 010, 110, 111, 101, 100, 000

Can be shown to be equivalent to:

000, 001, 011, 111, 101, 100, 110, 010

The above equivalence can be shown by the following steps of trivialoperations:

1) Negate the third column. This gives

000, 010, 011, 111, 110, 100, 101, 001

2) Swap the second and third columns. This gives

000, 001, 011, 111, 101, 100, 110, 010

The two labelings are considered equivalent because they yield the samePD Capacity as long as the constellation points locations are the same.

In the constellation point specifications shown in FIGS. 25-95, alabeling can be interchanged by any equivalent labeling withoutaffecting the performance parameters. A labeling used in thespecifications may not directly appear to be a cyclically rotated binaryreflective gray labeling, but it can be shown to be equivalent to one ormore cyclically rotated binary reflective gray labelings.

Prior Art Geometric Constellations

Geometric constellations have been specified in the prior art inattempts to achieve performance gains relative to constellations thatmaximize d_(min). Examples of such constellations are disclosed inSommer and Fettweis, “Signal Shaping by Non-Uniform QAM for AWGNChannels and Applications Using Turbo Coding” ITG Conference Source andChannel Coding, p. 81-86, 2000. The specific constellations disclosed bySommer and Fettweis for PAM-8, PAM-16, and PAM-32 are as follows:

PAM-8:

−1.6630 −0.9617 −0.5298 −0.1705 0.1705 0.5298 0.9617 1.6630PAM-16:

−1.9382 −1.3714 −1.0509 −0.8079 −0.6026 −0.4185 −0.2468 −0.0816 0.08160.2468 0.4185 0.6026 0.8079 1.0509 1.3714 1.9382PAM-32:

−2.1970 −1.7095 −1.4462 −1.2545 −1.0991 −0.9657 −0.8471 −0.7390 −0.6386−0.5441 −0.4540 −0.3673 −0.2832 −0.2010 −0.1201 −0.0400 0.0400 0.12010.2010 0.2832 0.3673 0.4540 0.5441 0.6386 0.7390 0.8471 0.9657 1.09911.2545 1.4462 1.7095 2.1970

Another class of geometric constellations is disclosed in Long et al.,“Approaching the AWGN Channel Capacity without Active Shaping”Proceedings of International Symposium on Information Theory, p. 374,1997. The specific PAM-8, PAM-16, and PAM-32 constellations disclosed byLong et al. are as follows:

PAM-8:

−3 −1 −1 −1 1 1 1 3PAM-16:

−4 −2 −2 −2 −2 0 0 0 0 0 0 2 2 2 2 4PAM-32:

−5 −3 −3 −3 −3 −3 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 1 1 3 33 3 3 5

The above prior art constellations are geometric and can provideperformance improvements at some SNRs relative to constellations thatmaximize d_(min). The performance of the constellations varies with SNRand at certain SNRs the constellations are proximate to capacityoptimized constellations. Therefore, the ranges specified in FIGS. 25-95are defined so that prior art constellations are excluded at thespecific SNRs at which these constellations are proximate to a capacityoptimized constellation.

Constructing Multidimensional Constellations

The tables shown in FIGS. 25-95 can be used to identify optimalN-dimensional constellations. The optimized multi-dimensionalconstellation can be determined by finding the Cartesian power X^(n) andthe resulting labeling constructed by finding the correspondingCartesian power of L^(n). Ranges within which the multi-dimensionalconstellation points can be selected (i.e. perturbed), can then bedefined with respect to each constellation point of the constructedmulti-dimensional constellation, using an n-dimensional perturbationvector, such that each component of the perturbation vector has amagnitude that is less than r_(max) defined by the range tables.

Example of a QAM Constellation

The optimized constellation points for a PAM-8 constellation optimizedfor PD capacity at SNR=9 dB are as follows:

−7.3992 −4.8128 −1.3438 −2.0696 7.3991 4.8129 1.3438 2.0691

The labelings corresponding to the above PAM-8 constellation points are:

000 001 010 011 100 101 110 111

Using this PAM-8 constellation, it is possible to construct a QAM-64constellation. While PAM-8 maps 3 bits to one dimension, QAM-64 maps 6bits to two dimensions. The first three bits will determine the locationin the X-dimension and the second three bits will determine the locationin the Y-dimension. The resulting QAM-64 constellation for example willmap the bits 000 010 to the two dimensional constellation point(−7.3992, −1.3438), and 111 110 to the two dimensional constellationpoint (2.0691, 1.3438). The points corresponding to the remaining labelscan be derived in a similar manner.

The ranges shown in FIGS. 25-95 can be utilized to select QAMconstellations in a similar manner to that outlined above with respectto the selection of a PAM-8 constellation based upon ranges specifiedwith respect to a PAM-8 constellation optimized for PD capacity at 9 dB.A range of 0.282 can be applied to every component of each twodimensional constellation point (based on the 50% of the achievable gainin capacity column). For example, the two points (−7.4992, −1.1438) and(2.2691, 1.1438) are within the ranges as they are spaced distances(−0.1, 0.2) and (0.2, −0.2) respectively from the optimizedconstellation points. In this way, the ranges can be used to identifyconstellations that are probabilistically likely to result in aperformance improvement relative to a constellation that maximizesd_(min).

The same procedure can apply to a constellation optimized for jointcapacity.

However, the choice of labeling does not affect joint capacity. Theabove procedure can similarly be applied to an N-dimensionalconstellation constructed from a PAM constellation.

Although the present invention has been described in certain specificembodiments, many additional modifications and variations would beapparent to those skilled in the art. It is therefore to be understoodthat the present invention may be practiced otherwise than specificallydescribed, without departing from the scope and spirit of the presentinvention. Thus, embodiments of the present invention should beconsidered in all respects as illustrative and not restrictive.

What is claimed is:
 1. A digital communication system, comprising: atransmitter configured to transmit signals to a receiver via acommunication channel; wherein the transmitter, comprises: a coderconfigured to receive user bits and output encoded bits at an expandedoutput encoded bit rate; a mapper configured to map encoded bits tosymbols in a symbol constellation; a modulator configured to generate asignal for transmission via the communication channel using symbolsgenerated by the mapper; wherein the receiver, comprises: a demodulatorconfigured to demodulate the received signal via the communicationchannel; a demapper configured to estimate likelihoods from the ademodulated signal; a decoder that is configured to estimate decodedbits from the likelihoods generated by the demapper; and wherein thesymbol constellation is a non-uniform QAM constellation formed byorthogonalizing a first PAM-32 symbol constellation and a second PAM-32symbol constellation, wherein the first PAM-32 symbol constellationachieves at least 50% of the gain in parallel decode capacity at a givenSNR achieved by a non-uniform PAM-32 symbol constellation optimized forparallel decode capacity at the given SNR relative to a uniform PAM-32symbol constellation.
 2. The digital communication system of claim 1,wherein the first PAM-32 symbol constellation and the second PAM-32symbol constellation are identical.
 3. The digital communication systemof claim 1, wherein the first PAM-32 symbol constellation comprises aset of constellation points where each point is within 6.1114×10⁻² fromthe following constellation points: −1.4547, −1.1665, −0.9897, −1.0006,−0.6848, −0.6937, −0.7234, −0.7234, −0.1871, −0.1876, −0.1891, −0.1891,−0.3177, −0.3177, −0.3177, −0.3177, 1.4547, 1.1665, 0.9897, 1.0006,0.6848, 0.6937, 0.7234, 0.7234, 0.1871, 0.1877, 0.1891, 0.1891, 0.3177,0.3177, 0.3177, 0.3177.
 4. The digital communication system of claim 1,wherein the first PAM-32 symbol constellation achieves at least 90% ofthe gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis within 2.6307×10⁻² from the following constellation points: −1.4547,−1.1665, −0.9897, −1.0006, −0.6848, −0.6937, −0.7234, −0.7234, −0.1871,−0.1876, −0.1891, −0.1891, −0.3177, −0.3177, −0.3177, −0.3177, 1.4547,1.1665, 0.9897, 1.0006, 0.6848, 0.6937, 0.7234, 0.7234, 0.1871, 0.1877,0.1891, 0.1891, 0.3177, 0.3177, 0.3177, 0.3177.
 5. The digitalcommunication system of claim 1, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.454, −1.1665, −0.9897, −1.0006, −0.6848,−0.6937, −0.7234, −0.7234, −0.1871, −0.1876, −0.1891, −0.1891, −0.3177,−0.3177, −0.3177, −0.3177, 1.4547, 1.1665, 0.9897, 1.0006, 0.6848,0.6937, 0.7234, 0.7234, 0.1871, 0.1877, 0.1891, 0.1891, 0.3177, 0.3177,0.3177, 0.3177.
 6. The digital communication system of claim 1, whereinthe first PAM-32 symbol constellation comprises a set of constellationpoints where each point is within 4.9358×10⁻² from the followingconstellation points: −1.4861, −1.2153, −0.9597, −1.0028, −0.6382,−0.6391, −0.7071, −0.7071, −0.1438, −0.1438, −0.1438, −0.1438, −0.3527,−0.3527, −0.3465, −0.3480, 1.4865, 1.2153, 0.9597, 1.0028, 0.6382,0.6390, 0.7071, 0.7071, 0.1437, 0.1437, 0.1437, 0.1437, 0.3527, 0.3527,0.3465, 0.3466.
 7. The digital communication system of claim 1, whereinthe first PAM-32 symbol constellation achieves at least 90% of the gainin parallel decode capacity of the given parallel decode capacityachieved by a non-uniform PAM-32 symbol constellation optimized forparallel decode capacity at the given SNR relative to a uniform PAM-32symbol constellation, and wherein the first PAM-32 symbol constellationcomprises a set of constellation points where each point is within2.3052×10⁻² from the following constellation points: −1.4861, −1.2153,−0.9597, −1.0028, −0.6382, −0.6391, −0.7071, −0.7071, −0.1438, −0.1438,−0.1438, −0.1438, −0.3527, −0.3527, −0.3465, −0.3480, 1.4865, 1.2153,0.9597, 1.0028, 0.6382, 0.6390, 0.7071, 0.7071, 0.1437, 0.1437, 0.1437,0.1437, 0.3527, 0.3527, 0.3465, 0.3466.
 8. The digital communicationsystem of claim 1, wherein the first PAM-32 symbol constellationachieves 100% of the gain in parallel decode capacity of the givenparallel decode capacity achieved by a non-uniform PAM-32 symbolconstellation optimized for parallel decode capacity at the given SNRrelative to a uniform PAM-32 symbol constellation, and wherein the firstPAM-32 symbol constellation comprises the following set of constellationpoints: −1.4861, −1.2153, −0.9597, −1.0028, −0.6382, −0.6391, −0.7071,−0.7071, −0.1438, −0.1438, −0.1438, −0.1438, −0.3527, −0.3527, −0.3465,−0.3480, 1.4865, 1.2153, 0.9597, 1.0028, 0.6382, 0.6390, 0.7071, 0.7071,0.1437, 0.1437, 0.1437, 0.1437, 0.3527, 0.3527, 0.3465, 0.3466.
 9. Thedigital communication system of claim 1, wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis within 4.7674×10⁻² from the following constellation points: −1.5023,−1.2278, −0.9434, −1.0038, −0.6110, −0.6110, −0.7176, −0.7108, −0.1288,−0.1288, −0.1301, −0.1301, −0.3659, −0.3659, −0.3439, −0.3441, 1.5024,1.2278, 0.9433, 1.0037, 0.6109, 0.6109, 0.7175, 0.7108, 0.1288, 0.1289,0.1301, 0.1301, 0.3659, 0.3659, 0.3439, 0.3441.
 10. The digitalcommunication system of claim 1, wherein the first PAM-32 symbolconstellation achieves at least 90% of the gain in parallel decodecapacity of the given parallel decode capacity achieved by a non-uniformPAM-32 symbol constellation optimized for parallel decode capacity atthe given SNR relative to a uniform PAM-32 symbol constellation, andwherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is 1.9874×10⁻² from the followingconstellation points: −1.5023, −1.2278, −0.9434, −1.0038, −0.6110,−0.6110, −0.7176, −0.7108, −0.1288, −0.1288, −0.1301, −0.1301, −0.3659,−0.3659, −0.3439, −0.3441, 1.5024, 1.2278, 0.9433, 1.0037, 0.6109,0.6109, 0.7175, 0.7108, 0.1288, 0.1289, 0.1301, 0.1301, 0.3659, 0.3659,0.3439, 0.3441.
 11. The digital communication system of claim 1, whereinthe first PAM-32 symbol constellation achieves 100% of the gain inparallel decode capacity of the given parallel decode capacity achievedby a non-uniform PAM-32 symbol constellation optimized for paralleldecode capacity at the given SNR relative to a uniform PAM-32 symbolconstellation, and wherein the first PAM-32 symbol constellationcomprises the following set of constellation points: −1.5023, −1.2278,−0.9434, −1.0038, −0.6110, −0.6110, −0.7176, −0.7108, −0.1288, −0.1288,−0.1301, −0.1301, −0.3659, −0.3659, −0.3439, −0.3441, 1.5024, 1.2278,0.9433, 1.0037, 0.6109, 0.6109, 0.7175, 0.7108, 0.1288, 0.1289, 0.1301,0.1301, 0.3659, 0.3659, 0.3439, 0.3441.
 12. The digital communicationsystem of claim 1, wherein the first PAM-32 symbol constellationcomprises a set of constellation points where each point is within3.8751×10⁻² from the following constellation points: −1.5003, −1.2326,−0.9288, −1.0219, −0.5860, −0.5860, −0.7430, −0.7215, −0.1015, −0.1016,−0.1366, −0.1366, −0.3901, −0.3901, −0.3167, −0.3170, 1.5005, 1.2327,0.9289, 1.0219, 0.5860, 0.5860, 0.7429, 0.7216, 0.1015, 0.1015, 0.1366,0.1366, 0.3900, 0.3900, 0.3167, 0.3169.
 13. The digital communicationsystem of claim 1, wherein the first PAM-32 symbol constellationachieves at least 90% of the gain in parallel decode capacity of thegiven parallel decode capacity achieved by a non-uniform PAM-32 symbolconstellation optimized for parallel decode capacity at the given SNRrelative to a uniform PAM-32 symbol constellation, and wherein the firstPAM-32 symbol constellation comprises a set of constellation pointswhere each point is 1.6963×10⁻² from the following constellation points:−1.5003, −1.2326, −0.9288, −1.0219, −0.5860, −0.5860, −0.7430, −0.7215,−0.1015, −0.1016, −0.1366, −0.1366, −0.3901, −0.3901, −0.3167, −0.3170,1.5005, 1.2327, 0.9289, 1.0219, 0.5860, 0.5860, 0.7429, 0.7216, 0.1015,0.1015, 0.1366, 0.1366, 0.3900, 0.3900, 0.3167, 0.3169.
 14. The digitalcommunication system of claim 1, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.5003, −1.2326, −0.9288, −1.0219, −0.5860,−0.5860, −0.7430, −0.7215, −0.1015, −0.1016, −0.1366, −0.1366, −0.3901,−0.3901, −0.3167, −0.3170, 1.5005, 1.2327, 0.9289, 1.0219, 0.5860,0.5860, 0.7429, 0.7216, 0.1015, 0.1015, 0.1366, 0.1366, 0.3900, 0.3900,0.3167, 0.3169.
 15. The digital communication system of claim 1, whereinthe first PAM-32 symbol constellation comprises a set of constellationpoints where each point is within 2.9715×10⁻² from the followingconstellation points: −1.4779, −1.2277, −0.9207, −1.0362, −0.5802,−0.5878, −0.7625, −0.7207, −0.0757, −0.0757, −0.1632, −0.1632, −0.4185,−0.4177, −0.3083, −0.3083, 1.4779, 1.2276, 0.9208, 1.0361, 0.5802,0.5878, 0.7625, 0.7207, 0.0758, 0.0758, 0.1632, 0.1632, 0.4184, 0.4177,0.3083, 0.3083.
 16. The digital communication system of claim 1, whereinthe first PAM-32 symbol constellation achieves at least 90% of the gainin parallel decode capacity of the given parallel decode capacityachieved by a non-uniform PAM-32 symbol constellation optimized forparallel decode capacity at the given SNR relative to a uniform PAM-32symbol constellation, and wherein the first PAM-32 symbol constellationcomprises a set of constellation points where each point is 1.3058×10⁻²from the following constellation points: −1.4779, −1.2277, −0.9207,−1.0362, −0.5802, −0.5878, −0.7625, −0.7207, −0.0757, −0.0757, −0.1632,−0.1632, −0.4185, −0.4177, −0.3083, −0.3083, 1.4779, 1.2276, 0.9208,1.0361, 0.5802, 0.5878, 0.7625, 0.7207, 0.0758, 0.0758, 0.1632, 0.1632,0.4184, 0.4177, 0.3083, 0.3083.
 17. The digital communication system ofclaim 1, wherein the first PAM-32 symbol constellation achieves 100% ofthe gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises the following set of constellation points:−1.4779, −1.2277, −0.9207, −1.0362, −0.5802, −0.5878, −0.7625, −0.7207,−0.0757, −0.0757, −0.1632, −0.1632, −0.4185, −0.4177, −0.3083, −0.3083,1.4779, 1.2276, 0.9208, 1.0361, 0.5802, 0.5878, 0.7625, 0.7207, 0.0758,0.0758, 0.1632, 0.1632, 0.4184, 0.4177, 0.3083, 0.3083.
 18. The digitalcommunication system of claim 1, wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis within 2.8949×10⁻² from the following constellation points: −1.4700,−1.2271, −0.9199, −1.0418, −0.5753, −0.5899, −0.7720, −0.7166, −0.0710,−0.0710, −0.1686, −0.1686, −0.4245, −0.4221, −0.3073, −0.3074, 1.4677,1.2269, 0.9198, 1.0416, 0.5754, 0.5899, 0.7719, 0.7166, 0.0711, 0.0711,0.1687, 0.1687, 0.4246, 0.4222, 0.3074, 0.3075.
 19. The digitalcommunication system of claim 1, wherein the first PAM-32 symbolconstellation achieves at least 90% of the gain in parallel decodecapacity of the given parallel decode capacity achieved by a non-uniformPAM-32 symbol constellation optimized for parallel decode capacity atthe given SNR relative to a uniform PAM-32 symbol constellation, andwherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is 1.1871×10⁻² from the followingconstellation points: −1.4700, −1.2271, −0.9199, −1.0418, −0.5753,−0.5899, −0.7720, −0.7166, −0.0710, −0.0710, −0.1686, −0.1686, −0.4245,−0.4221, −0.3073, −0.3074, 1.4677, 1.2269, 0.9198, 1.0416, 0.5754,0.5899, 0.7719, 0.7166, 0.0711, 0.0711, 0.1687, 0.1687, 0.4246, 0.4222,0.3074, 0.3075.
 20. The digital communication system of claim 1, whereinthe first PAM-32 symbol constellation achieves 100% of the gain inparallel decode capacity of the given parallel decode capacity achievedby a non-uniform PAM-32 symbol constellation optimized for paralleldecode capacity at the given SNR relative to a uniform PAM-32 symbolconstellation, and wherein the first PAM-32 symbol constellationcomprises the following set of constellation points: −1.4700, −1.2271,−0.9199, −1.0418, −0.5753, −0.5899, −0.7720, −0.7166, −0.0710, −0.0710,−0.1686, −0.1686, −0.4245, −0.4221, −0.3073, −0.3074, 1.4677, 1.2269,0.9198, 1.0416, 0.5754, 0.5899, 0.7719, 0.7166, 0.0711, 0.0711, 0.1687,0.1687, 0.4246, 0.4222, 0.3074, 0.3075.
 21. The digital communicationsystem of claim 1, wherein the first PAM-32 symbol constellationcomprises a set of constellation points where each point is within2.0371×10⁻² from the following constellation points: −1.4459, −1.2257,−0.9275, −1.0560, −0.5517, −0.6082, −0.8006, −0.7125, −0.0628, −0.0637,−0.1743, −0.1712, −0.4396, −0.4097, −0.2947, −0.3057, 1.4457, 1.2256,0.9274, 1.0560, 0.5516, 0.6082, 0.8005, 0.7125, 0.0629, 0.0638, 0.1744,0.1713, 0.4396, 0.4098, 0.2949, 0.3058.
 22. The digital communicationsystem of claim 1, wherein the first PAM-32 symbol constellationachieves at least 90% of the gain in parallel decode capacity of thegiven parallel decode capacity achieved by a non-uniform PAM-32 symbolconstellation optimized for parallel decode capacity at the given SNRrelative to a uniform PAM-32 symbol constellation, and wherein the firstPAM-32 symbol constellation comprises a set of constellation pointswhere each point is 0.9726×10⁻² from the following constellation points:−1.4459, −1.2257, −0.9275, −1.0560, −0.5517, −0.6082, −0.8006, −0.7125,−0.0628, −0.0637, −0.1743, −0.1712, −0.4396, −0.4097, −0.2947, −0.3057,1.4457, 1.2256, 0.9274, 1.0560, 0.5516, 0.6082, 0.8005, 0.7125, 0.0629,0.0638, 0.1744, 0.1713, 0.4396, 0.4098, 0.2949, 0.3058.
 23. The digitalcommunication system of claim 1, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.4459, −1.2257, −0.9275, −1.0560, −0.5517,−0.6082, −0.8006, −0.7125, −0.0628, −0.0637, −0.1743, −0.1712, −0.4396,−0.4097, −0.2947, −0.3057, 1.4457, 1.2256, 0.9274, 1.0560, 0.5516,0.6082, 0.8005, 0.7125, 0.0629, 0.0638, 0.1744, 0.1713, 0.4396, 0.4098,0.2949, 0.3058.
 24. The digital communication system of claim 1, whereinthe first PAM-32 symbol constellation comprises a set of constellationpoints where each point is within 1.6887×10⁻² from the followingconstellation points: −1.4100, −1.2114, −0.9340, −1.0567, −0.5593,−0.6315, −0.8180, −0.7261, −0.0437, −0.0791, −0.1969, −0.1577, −0.4687,−0.4099, −0.2822, −0.3293, 1.4100, 1.2114, 0.9340, 1.0567, 0.5593,0.6315, 0.8180, 0.7261, 0.0436, 0.0791, 0.1969, 0.1577, 0.4687, 0.4099,0.2822, 0.3292.
 25. The digital communication system of claim 1, whereinthe first PAM-32 symbol constellation achieves at least 90% of the gainin parallel decode capacity of the given parallel decode capacityachieved by a non-uniform PAM-32 symbol constellation optimized forparallel decode capacity at the given SNR relative to a uniform PAM-32symbol constellation, and wherein the first PAM-32 symbol constellationcomprises a set of constellation points where each point is 0.7390×10⁻²from the following constellation points: −1.4100, −1.2114, −0.9340,−1.0567, −0.5593, −0.6315, −0.8180, −0.7261, −0.0437, −0.0791, −0.1969,−0.1577, −0.4687, −0.4099, −0.2822, −0.3293, 1.4100, 1.2114, 0.9340,1.0567, 0.5593, 0.6315, 0.8180, 0.7261, 0.0436, 0.0791, 0.1969, 0.1577,0.4687, 0.4099, 0.2822, 0.3292.
 26. The digital communication system ofclaim 1, wherein the first PAM-32 symbol constellation achieves 100% ofthe gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises the following set of constellation points:−1.4100, −1.2114, −0.9340, −1.0567, −0.5593, −0.6315, −0.8180, −0.7261,−0.0437, −0.0791, −0.1969, −0.1577, −0.4687, −0.4099, −0.2822, −0.3293,1.4100, 1.2114, 0.9340, 1.0567, 0.5593, 0.6315, 0.8180, 0.7261, 0.0436,0.0791, 0.1969, 0.1577, 0.4687, 0.4099, 0.2822, 0.3292.
 27. The digitalcommunication system of claim 1, wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis within 1.0837×10⁻² from the following constellation points: −1.3701,−1.1930, −0.9383, −1.0532, −0.5751, −0.6518, −0.8316, −0.7407, −0.0357,−0.0916, −0.2175, −0.1599, −0.4931, −0.4252, −0.2903, −0.3516, 1.3701,1.1930, 0.9383, 1.0532, 0.5751, 0.6518, 0.8316, 0.7407, 0.0357, 0.0916,0.2175, 0.1599, 0.4931, 0.4252, 0.2903, 0.3516.
 28. The digitalcommunication system of claim 1, wherein the first PAM-32 symbolconstellation achieves at least 90% of the gain in parallel decodecapacity of the given parallel decode capacity achieved by a non-uniformPAM-32 symbol constellation optimized for parallel decode capacity atthe given SNR relative to a uniform PAM-32 symbol constellation, andwherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is 0.4672×10⁻² from the followingconstellation points: −1.3701, −1.1930, −0.9383, −1.0532, −0.5751,−0.6518, −0.8316, −0.7407, −0.0357, −0.0916, −0.2175, −0.1599, −0.4931,−0.4252, −0.2903, −0.3516, 1.3701, 1.1930, 0.9383, 1.0532, 0.5751,0.6518, 0.8316, 0.7407, 0.0357, 0.0916, 0.2175, 0.1599, 0.4931, 0.4252,0.2903, 0.3516.
 29. The digital communication system of claim 1, whereinthe first PAM-32 symbol constellation achieves 100% of the gain inparallel decode capacity of the given parallel decode capacity achievedby a non-uniform PAM-32 symbol constellation optimized for paralleldecode capacity at the given SNR relative to a uniform PAM-32 symbolconstellation, and wherein the first PAM-32 symbol constellationcomprises the following set of constellation points: −1.3701, −1.1930,−0.9383, −1.0532, −0.5751, −0.6518, −0.8316, −0.7407, −0.0357, −0.0916,−0.2175, −0.1599, −0.4931, −0.4252, −0.2903, −0.3516, 1.3701, 1.1930,0.9383, 1.0532, 0.5751, 0.6518, 0.8316, 0.7407, 0.0357, 0.0916, 0.2175,0.1599, 0.4931, 0.4252, 0.2903, 0.3516.
 30. The digital communicationsystem of claim 1, wherein the first PAM-32 symbol constellationcomprises a set of constellation points where each point is within0.7084×10⁻² from the following constellation points: −1.3295, −1.1740,−0.9426, −1.0490, −0.5927, −0.6709, −0.8440, −0.7558, −0.0342, −0.0985,−0.2317, −0.1663, −0.5143, −0.4424, −0.3021, −0.3699, 1.3295, 1.1739,0.9426, 1.0490, 0.5927, 0.6709, 0.8440, 0.7558, 0.0342, 0.0985, 0.2317,0.1663, 0.5144, 0.4424, 0.3021, 0.3664.
 31. The digital communicationsystem of claim 1, wherein the first PAM-32 symbol constellationachieves at least 90% of the gain in parallel decode capacity of thegiven parallel decode capacity achieved by a non-uniform PAM-32 symbolconstellation optimized for parallel decode capacity at the given SNRrelative to a uniform PAM-32 symbol constellation, and wherein the firstPAM-32 symbol constellation comprises a set of constellation pointswhere each point is 0.3063×10⁻² from the following constellation points:−1.3295, −1.1740, −0.9426, −1.0490, −0.5927, −0.6709, −0.8440, −0.7558,−0.0342, −0.0985, −0.2317, −0.1663, −0.5143, −0.4424, −0.3021, −0.3699,1.3295, 1.1739, 0.9426, 1.0490, 0.5927, 0.6709, 0.8440, 0.7558, 0.0342,0.0985, 0.2317, 0.1663, 0.5144, 0.4424, 0.3021, 0.3664.
 32. The digitalcommunication system of claim 1, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.3295, −1.1740, −0.9426, −1.0490, −0.5927,−0.6709, −0.8440, −0.7558, −0.0342, −0.0985, −0.2317, −0.1663, −0.5143,−0.4424, −0.3021, −0.3699, 1.3295, 1.1739, 0.9426, 1.0490, 0.5927,0.6709, 0.8440, 0.7558, 0.0342, 0.0985, 0.2317, 0.1663, 0.5144, 0.4424,0.3021, 0.3664.
 33. The digital communication system of claim 1, whereinthe first PAM-32 symbol constellation achieves at least 90% of the gainin parallel decode capacity of the given parallel decode capacityachieved by a non-uniform PAM-32 symbol constellation optimized forparallel decode capacity at the given SNR relative to a uniform PAM-32symbol constellation, and wherein the first PAM-32 symbol constellationcomprises a set of constellation points where each point is 0.3063×10⁻²from the following constellation points: −1.3295, −1.1740, −0.9426,−1.0490, −0.5927, −0.6709, −0.8440, −0.7558, −0.0342, −0.0985, −0.2317,−0.1663, −0.5143, −0.4424, −0.3021, −0.3699, 1.3295, 1.1739, 0.9426,1.0490, 0.5927, 0.6709, 0.8440, 0.7558, 0.0342, 0.0985, 0.2317, 0.1663,0.5144, 0.4424, 0.3021, 0.3664.
 34. A digital communication system,comprising: a transmitter configured to transmit signals via acommunication channel, wherein the transmitter, comprises: a coderconfigured to receive user bits and output encoded bits at an expandedoutput encoded bit rate; a mapper configured to map encoded bits tosymbols in a symbol constellation; a modulator configured to generate asignal for transmission via the communication channel using symbolsgenerated by the mapper; wherein the symbol constellation is anon-uniform QAM constellation formed by orthogonalizing a first PAM-32symbol constellation and a second PAM-32 symbol constellation, whereinthe first PAM-32 symbol constellation achieves at least 50% of the gainin parallel decode capacity at a given SNR achieved by a non-uniformPAM-32 symbol constellation optimized for parallel decode capacity atthe given SNR relative to a uniform PAM-32 symbol constellation.
 35. Thedigital communication system of claim 34, wherein the first PAM-32symbol constellation and the second PAM-32 symbol constellation areidentical.
 36. The digital communication system of claim 34, wherein thefirst PAM-32 symbol constellation comprises a set of constellationpoints where each point is within 6.1114×10⁻² from the followingconstellation points: −1.4547, −1.1665, −0.9897, −1.0006, −0.6848,−0.6937, −0.7234, −0.7234, −0.1871, −0.1876, −0.1891, −0.1891, −0.3177,−0.3177, −0.3177, −0.3177, 1.4547, 1.1665, 0.9897, 1.0006, 0.6848,0.6937, 0.7234, 0.7234, 0.1871, 0.1877, 0.1891, 0.1891, 0.3177, 0.3177,0.3177, 0.3177.
 37. The digital communication system of claim 34,wherein the first PAM-32 symbol constellation achieves at least 90% ofthe gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis within 2.6307×10⁻² from the following constellation points: −1.4547,−1.1665, −0.9897, −1.0006, −0.6848, −0.6937, −0.7234, −0.7234, −0.1871,−0.1876, −0.1891, −0.1891, −0.3177, −0.3177, −0.3177, −0.3177, 1.4547,1.1665, 0.9897, 1.0006, 0.6848, 0.6937, 0.7234, 0.7234, 0.1871, 0.1877,0.1891, 0.1891, 0.3177, 0.3177, 0.3177, 0.3177.
 38. The digitalcommunication system of claim 34, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.454, −1.1665, −0.9897, −1.0006, −0.6848,−0.6937, −0.7234, −0.7234, −0.1871, −0.1876, −0.1891, −0.1891, −0.3177,−0.3177, −0.3177, −0.3177, 1.4547, 1.1665, 0.9897, 1.0006, 0.6848,0.6937, 0.7234, 0.7234, 0.1871, 0.1877, 0.1891, 0.1891, 0.3177, 0.3177,0.3177, 0.3177.
 39. The digital communication system of claim 34,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 4.9358×10⁻² from thefollowing constellation points: −1.4861, −1.2153, −0.9597, −1.0028,−0.6382, −0.6391, −0.7071, −0.7071, −0.1438, −0.1438, −0.1438, −0.1438,−0.3527, −0.3527, −0.3465, −0.3480, 1.4865, 1.2153, 0.9597, 1.0028,0.6382, 0.6390, 0.7071, 0.7071, 0.1437, 0.1437, 0.1437, 0.1437, 0.3527,0.3527, 0.3465, 0.3466.
 40. The digital communication system of claim34, wherein the first PAM-32 symbol constellation achieves at least 90%of the gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis within 2.3052×10⁻² from the following constellation points: −1.4861,−1.2153, −0.9597, −1.0028, −0.6382, −0.6391, −0.7071, −0.7071, −0.1438,−0.1438, −0.1438, −0.1438, −0.3527, −0.3527, −0.3465, −0.3480, 1.4865,1.2153, 0.9597, 1.0028, 0.6382, 0.6390, 0.7071, 0.7071, 0.1437, 0.1437,0.1437, 0.1437, 0.3527, 0.3527, 0.3465, 0.3466.
 41. The digitalcommunication system of claim 34, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.4861, −1.2153, −0.9597, −1.0028, −0.6382,−0.6391, −0.7071, −0.7071, −0.1438, −0.1438, −0.1438, −0.1438, −0.3527,−0.3527, −0.3465, −0.3480, 1.4865, 1.2153, 0.9597, 1.0028, 0.6382,0.6390, 0.7071, 0.7071, 0.1437, 0.1437, 0.1437, 0.1437, 0.3527, 0.3527,0.3465, 0.3466.
 42. The digital communication system of claim 34,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 4.7674×10⁻² from thefollowing constellation points: −1.5023, −1.2278, −0.9434, −1.0038,−0.6110, −0.6110, −0.7176, −0.7108, −0.1288, −0.1288, −0.1301, −0.1301,−0.3659, −0.3659, −0.3439, −0.3441, 1.5024, 1.2278, 0.9433, 1.0037,0.6109, 0.6109, 0.7175, 0.7108, 0.1288, 0.1289, 0.1301, 0.1301, 0.3659,0.3659, 0.3439, 0.3441.
 43. The digital communication system of claim34, wherein the first PAM-32 symbol constellation achieves at least 90%of the gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis 1.9874×10⁻² from the following constellation points: −1.5023,−1.2278, −0.9434, −1.0038, −0.6110, −0.6110, −0.7176, −0.7108, −0.1288,−0.1288, −0.1301, −0.1301, −0.3659, −0.3659, −0.3439, −0.3441, 1.5024,1.2278, 0.9433, 1.0037, 0.6109, 0.6109, 0.7175, 0.7108, 0.1288, 0.1289,0.1301, 0.1301, 0.3659, 0.3659, 0.3439, 0.3441.
 44. The digitalcommunication system of claim 34, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.5023, −1.2278, −0.9434, −1.0038, −0.6110,−0.6110, −0.7176, −0.7108, −0.1288, −0.1288, −0.1301, −0.1301, −0.3659,−0.3659, −0.3439, −0.3441, 1.5024, 1.2278, 0.9433, 1.0037, 0.6109,0.6109, 0.7175, 0.7108, 0.1288, 0.1289, 0.1301, 0.1301, 0.3659, 0.3659,0.3439, 0.3441.
 45. The digital communication system of claim 34,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 3.8751×10⁻² from thefollowing constellation points: −1.5003, −1.2326, −0.9288, −1.0219,−0.5860, −0.5860, −0.7430, −0.7215, −0.1015, −0.1016, −0.1366, −0.1366,−0.3901, −0.3901, −0.3167, −0.3170, 1.5005, 1.2327, 0.9289, 1.0219,0.5860, 0.5860, 0.7429, 0.7216, 0.1015, 0.1015, 0.1366, 0.1366, 0.3900,0.3900, 0.3167, 0.3169.
 46. The digital communication system of claim34, wherein the first PAM-32 symbol constellation achieves at least 90%of the gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis 1.6963×10⁻² from the following constellation points: −1.5003,−1.2326, −0.9288, −1.0219, −0.5860, −0.5860, −0.7430, −0.7215, −0.1015,−0.1016, −0.1366, −0.1366, −0.3901, −0.3901, −0.3167, −0.3170, 1.5005,1.2327, 0.9289, 1.0219, 0.5860, 0.5860, 0.7429, 0.7216, 0.1015, 0.1015,0.1366, 0.1366, 0.3900, 0.3900, 0.3167, 0.3169.
 47. The digitalcommunication system of claim 34, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.5003, −1.2326, −0.9288, −1.0219, −0.5860,−0.5860, −0.7430, −0.7215, −0.1015, −0.1016, −0.1366, −0.1366, −0.3901,−0.3901, −0.3167, −0.3170, 1.5005, 1.2327, 0.9289, 1.0219, 0.5860,0.5860, 0.7429, 0.7216, 0.1015, 0.1015, 0.1366, 0.1366, 0.3900, 0.3900,0.3167, 0.3169.
 48. The digital communication system of claim 34,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 2.9715×10⁻² from thefollowing constellation points: −1.4779, −1.2277, −0.9207, −1.0362,−0.5802, −0.5878, −0.7625, −0.7207, −0.0757, −0.0757, −0.1632, −0.1632,−0.4185, −0.4177, −0.3083, −0.3083, 1.4779, 1.2276, 0.9208, 1.0361,0.5802, 0.5878, 0.7625, 0.7207, 0.0758, 0.0758, 0.1632, 0.1632, 0.4184,0.4177, 0.3083, 0.3083.
 49. The digital communication system of claim34, wherein the first PAM-32 symbol constellation achieves at least 90%of the gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis 1.3058×10⁻² from the following constellation points: −1.4779,−1.2277, −0.9207, −1.0362, −0.5802, −0.5878, −0.7625, −0.7207, −0.0757,−0.0757, −0.1632, −0.1632, −0.4185, −0.4177, −0.3083, −0.3083, 1.4779,1.2276, 0.9208, 1.0361, 0.5802, 0.5878, 0.7625, 0.7207, 0.0758, 0.0758,0.1632, 0.1632, 0.4184, 0.4177, 0.3083, 0.3083.
 50. The digitalcommunication system of claim 34, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.4779, −1.2277, −0.9207, −1.0362, −0.5802,−0.5878, −0.7625, −0.7207, −0.0757, −0.0757, −0.1632, −0.1632, −0.4185,−0.4177, −0.3083, −0.3083, 1.4779, 1.2276, 0.9208, 1.0361, 0.5802,0.5878, 0.7625, 0.7207, 0.0758, 0.0758, 0.1632, 0.1632, 0.4184, 0.4177,0.3083, 0.3083.
 51. The digital communication system of claim 34,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 2.8949×10⁻² from thefollowing constellation points: −1.4700, −1.2271, −0.9199, −1.0418,−0.5753, −0.5899, −0.7720, −0.7166, −0.0710, −0.0710, −0.1686, −0.1686,−0.4245, −0.4221, −0.3073, −0.3074, 1.4677, 1.2269, 0.9198, 1.0416,0.5754, 0.5899, 0.7719, 0.7166, 0.0711, 0.0711, 0.1687, 0.1687, 0.4246,0.4222, 0.3074, 0.3075.
 52. The digital communication system of claim34, wherein the first PAM-32 symbol constellation achieves at least 90%of the gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis 1.1871×10⁻² from the following constellation points: −1.4700,−1.2271, −0.9199, −1.0418, −0.5753, −0.5899, −0.7720, −0.7166, −0.0710,−0.0710, −0.1686, −0.1686, −0.4245, −0.4221, −0.3073, −0.3074, 1.4677,1.2269, 0.9198, 1.0416, 0.5754, 0.5899, 0.7719, 0.7166, 0.0711, 0.0711,0.1687, 0.1687, 0.4246, 0.4222, 0.3074, 0.3075.
 53. The digitalcommunication system of claim 34, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.4700, −1.2271, −0.9199, −1.0418, −0.5753,−0.5899, −0.7720, −0.7166, −0.0710, −0.0710, −0.1686, −0.1686, −0.4245,−0.4221, −0.3073, −0.3074, 1.4677, 1.2269, 0.9198, 1.0416, 0.5754,0.5899, 0.7719, 0.7166, 0.0711, 0.0711, 0.1687, 0.1687, 0.4246, 0.4222,0.3074, 0.3075.
 54. The digital communication system of claim 34,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 2.0371×10⁻² from thefollowing constellation points: −1.4459, −1.2257, −0.9275, −1.0560,−0.5517, −0.6082, −0.8006, −0.7125, −0.0628, −0.0637, −0.1743, −0.1712,−0.4396, −0.4097, −0.2947, −0.3057, 1.4457, 1.2256, 0.9274, 1.0560,0.5516, 0.6082, 0.8005, 0.7125, 0.0629, 0.0638, 0.1744, 0.1713, 0.4396,0.4098, 0.2949, 0.3058.
 55. The digital communication system of claim34, wherein the first PAM-32 symbol constellation achieves at least 90%of the gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis 0.9726×10⁻² from the following constellation points: −1.4459,−1.2257, −0.9275, −1.0560, −0.5517, −0.6082, −0.8006, −0.7125, −0.0628,−0.0637, −0.1743, −0.1712, −0.4396, −0.4097, −0.2947, −0.3057, 1.4457,1.2256, 0.9274, 1.0560, 0.5516, 0.6082, 0.8005, 0.7125, 0.0629, 0.0638,0.1744, 0.1713, 0.4396, 0.4098, 0.2949, 0.3058.
 56. The digitalcommunication system of claim 34, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.4459, −1.2257, −0.9275, −1.0560, −0.5517,−0.6082, −0.8006, −0.7125, −0.0628, −0.0637, −0.1743, −0.1712, −0.4396,−0.4097, −0.2947, −0.3057, 1.4457, 1.2256, 0.9274, 1.0560, 0.5516,0.6082, 0.8005, 0.7125, 0.0629, 0.0638, 0.1744, 0.1713, 0.4396, 0.4098,0.2949, 0.3058.
 57. The digital communication system of claim 34,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 1.6887×10⁻² from thefollowing constellation points: −1.4100, −1.2114, −0.9340, −1.0567,−0.5593, −0.6315, −0.8180, −0.7261, −0.0437, −0.0791, −0.1969, −0.1577,−0.4687, −0.4099, −0.2822, −0.3293, 1.4100, 1.2114, 0.9340, 1.0567,0.5593, 0.6315, 0.8180, 0.7261, 0.0436, 0.0791, 0.1969, 0.1577, 0.4687,0.4099, 0.2822, 0.3292.
 58. The digital communication system of claim34, wherein the first PAM-32 symbol constellation achieves at least 90%of the gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis 0.7390×10⁻² from the following constellation points: −1.4100,−1.2114, −0.9340, −1.0567, −0.5593, −0.6315, −0.8180, −0.7261, −0.0437,−0.0791, −0.1969, −0.1577, −0.4687, −0.4099, −0.2822, −0.3293, 1.4100,1.2114, 0.9340, 1.0567, 0.5593, 0.6315, 0.8180, 0.7261, 0.0436, 0.0791,0.1969, 0.1577, 0.4687, 0.4099, 0.2822, 0.3292.
 59. The digitalcommunication system of claim 34, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.4100, −1.2114, −0.9340, −1.0567, −0.5593,−0.6315, −0.8180, −0.7261, −0.0437, −0.0791, −0.1969, −0.1577, −0.4687,−0.4099, −0.2822, −0.3293, 1.4100, 1.2114, 0.9340, 1.0567, 0.5593,0.6315, 0.8180, 0.7261, 0.0436, 0.0791, 0.1969, 0.1577, 0.4687, 0.4099,0.2822, 0.3292.
 60. The digital communication system of claim 34,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 1.0837×10⁻² from thefollowing constellation points: −1.3701, −1.1930, −0.9383, −1.0532,−0.5751, −0.6518, −0.8316, −0.7407, −0.0357, −0.0916, −0.2175, −0.1599,−0.4931, −0.4252, −0.2903, −0.3516, 1.3701, 1.1930, 0.9383, 1.0532,0.5751, 0.6518, 0.8316, 0.7407, 0.0357, 0.0916, 0.2175, 0.1599, 0.4931,0.4252, 0.2903, 0.3516.
 61. The digital communication system of claim34, wherein the first PAM-32 symbol constellation achieves at least 90%of the gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis 0.4672×10⁻² from the following constellation points: −1.3701,−1.1930, −0.9383, −1.0532, −0.5751, −0.6518, −0.8316, −0.7407, −0.0357,−0.0916, −0.2175, −0.1599, −0.4931, −0.4252, −0.2903, −0.3516, 1.3701,1.1930, 0.9383, 1.0532, 0.5751, 0.6518, 0.8316, 0.7407, 0.0357, 0.0916,0.2175, 0.1599, 0.4931, 0.4252, 0.2903, 0.3516.
 62. The digitalcommunication system of claim 34, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.3701, −1.1930, −0.9383, −1.0532, −0.5751,−0.6518, −0.8316, −0.7407, −0.0357, −0.0916, −0.2175, −0.1599, −0.4931,−0.4252, −0.2903, −0.3516, 1.3701, 1.1930, 0.9383, 1.0532, 0.5751,0.6518, 0.8316, 0.7407, 0.0357, 0.0916, 0.2175, 0.1599, 0.4931, 0.4252,0.2903, 0.3516.
 63. The digital communication system of claim 34,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 0.7084×10⁻² from thefollowing constellation points: −1.3295, −1.1740, −0.9426, −1.0490,−0.5927, −0.6709, −0.8440, −0.7558, −0.0342, −0.0985, −0.2317, −0.1663,−0.5143, −0.4424, −0.3021, −0.3699, 1.3295, 1.1739, 0.9426, 1.0490,0.5927, 0.6709, 0.8440, 0.7558, 0.0342, 0.0985, 0.2317, 0.1663, 0.5144,0.4424, 0.3021, 0.3664.
 64. The digital communication system of claim34, wherein the first PAM-32 symbol constellation achieves 100% of thegain in parallel decode capacity of the given parallel decode capacityachieved by a non-uniform PAM-32 symbol constellation optimized forparallel decode capacity at the given SNR relative to a uniform PAM-32symbol constellation, and wherein the first PAM-32 symbol constellationcomprises the following set of constellation points: −1.3295, −1.1740,−0.9426, −1.0490, −0.5927, −0.6709, −0.8440, −0.7558, −0.0342, −0.0985,−0.2317, −0.1663, −0.5143, −0.4424, −0.3021, −0.3699, 1.3295, 1.1739,0.9426, 1.0490, 0.5927, 0.6709, 0.8440, 0.7558, 0.0342, 0.0985, 0.2317,0.1663, 0.5144, 0.4424, 0.3021, 0.3664.
 65. A digital communicationsystem, comprising: a receiver via a communication channel transmittedsignals via a communication channel, wherein the receiver, comprises: ademodulator configured to demodulate the received signal via thecommunication channel; a demapper configured to estimate likelihoodsfrom the a demodulated signal and a symbol constellation; and a decoderthat is configured to estimate decoded bits from the likelihoodsgenerated by the demapper; wherein the symbol constellation is anon-uniform QAM constellation formed by orthogonalizing a first PAM-32symbol constellation and a second PAM-32 symbol constellation, whereinthe first PAM-32 symbol constellation achieves at least 50% of the gainin parallel decode capacity at a given SNR achieved by a non-uniformPAM-32 symbol constellation optimized for parallel decode capacity atthe given SNR relative to a uniform PAM-32 symbol constellation.
 66. Thedigital communication system of claim 65, wherein the first PAM-32symbol constellation and the second PAM-32 symbol constellation areidentical.
 67. The digital communication system of claim 65, wherein thefirst PAM-32 symbol constellation comprises a set of constellationpoints where each point is within 6.1114×10⁻² from the followingconstellation points: −1.4547, −1.1665, −0.9897, −1.0006, −0.6848,−0.6937, −0.7234, −0.7234, −0.1871, −0.1876, −0.1891, −0.1891, −0.3177,−0.3177, −0.3177, −0.3177, 1.4547, 1.1665, 0.9897, 1.0006, 0.6848,0.6937, 0.7234, 0.7234, 0.1871, 0.1877, 0.1891, 0.1891, 0.3177, 0.3177,0.3177, 0.3177.
 68. The digital communication system of claim 65,wherein the first PAM-32 symbol constellation achieves at least 90% ofthe gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis within 2.6307×10⁻² from the following constellation points: −1.4547,−1.1665, −0.9897, −1.0006, −0.6848, −0.6937, −0.7234, −0.7234, −0.1871,−0.1876, −0.1891, −0.1891, −0.3177, −0.3177, −0.3177, −0.3177, 1.4547,1.1665, 0.9897, 1.0006, 0.6848, 0.6937, 0.7234, 0.7234, 0.1871, 0.1877,0.1891, 0.1891, 0.3177, 0.3177, 0.3177, 0.3177.
 69. The digitalcommunication system of claim 65, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.454, −1.1665, −0.9897, −1.0006, −0.6848,−0.6937, −0.7234, −0.7234, −0.1871, −0.1876, −0.1891, −0.1891, −0.3177,−0.3177, −0.3177, −0.3177, 1.4547, 1.1665, 0.9897, 1.0006, 0.6848,0.6937, 0.7234, 0.7234, 0.1871, 0.1877, 0.1891, 0.1891, 0.3177, 0.3177,0.3177, 0.3177.
 70. The digital communication system of claim 65,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 4.9358×10⁻² from thefollowing constellation points: −1.4861, −1.2153, −0.9597, −1.0028,−0.6382, −0.6391, −0.7071, −0.7071, −0.1438, −0.1438, −0.1438, −0.1438,−0.3527, −0.3527, −0.3465, −0.3480, 1.4865, 1.2153, 0.9597, 1.0028,0.6382, 0.6390, 0.7071, 0.7071, 0.1437, 0.1437, 0.1437, 0.1437, 0.3527,0.3527, 0.3465, 0.3466.
 71. The digital communication system of claim65, wherein the first PAM-32 symbol constellation achieves at least 90%of the gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis within 2.3052×10⁻² from the following constellation points: −1.4861,−1.2153, −0.9597, −1.0028, −0.6382, −0.6391, −0.7071, −0.7071, −0.1438,−0.1438, −0.1438, −0.1438, −0.3527, −0.3527, −0.3465, −0.3480, 1.4865,1.2153, 0.9597, 1.0028, 0.6382, 0.6390, 0.7071, 0.7071, 0.1437, 0.1437,0.1437, 0.1437, 0.3527, 0.3527, 0.3465, 0.3466.
 72. The digitalcommunication system of claim 65, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.4861, −1.2153, −0.9597, −1.0028, −0.6382,−0.6391, −0.7071, −0.7071, −0.1438, −0.1438, −0.1438, −0.1438, −0.3527,−0.3527, −0.3465, −0.3480, 1.4865, 1.2153, 0.9597, 1.0028, 0.6382,0.6390, 0.7071, 0.7071, 0.1437, 0.1437, 0.1437, 0.1437, 0.3527, 0.3527,0.3465, 0.3466.
 73. The digital communication system of claim 65,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 4.7674×10⁻² from thefollowing constellation points: −1.5023, −1.2278, −0.9434, −1.0038,−0.6110, −0.6110, −0.7176, −0.7108, −0.1288, −0.1288, −0.1301, −0.1301,−0.3659, −0.3659, −0.3439, −0.3441, 1.5024, 1.2278, 0.9433, 1.0037,0.6109, 0.6109, 0.7175, 0.7108, 0.1288, 0.1289, 0.1301, 0.1301, 0.3659,0.3659, 0.3439, 0.3441.
 74. The digital communication system of claim65, wherein the first PAM-32 symbol constellation achieves at least 90%of the gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis 1.9874×10⁻² from the following constellation points: −1.5023,−1.2278, −0.9434, −1.0038, −0.6110, −0.6110, −0.7176, −0.7108, −0.1288,−0.1288, −0.1301, −0.1301, −0.3659, −0.3659, −0.3439, −0.3441, 1.5024,1.2278, 0.9433, 1.0037, 0.6109, 0.6109, 0.7175, 0.7108, 0.1288, 0.1289,0.1301, 0.1301, 0.3659, 0.3659, 0.3439, 0.3441.
 75. The digitalcommunication system of claim 65, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.5023, −1.2278, −0.9434, −1.0038, −0.6110,−0.6110, −0.7176, −0.7108, −0.1288, −0.1288, −0.1301, −0.1301, −0.3659,−0.3659, −0.3439, −0.3441, 1.5024, 1.2278, 0.9433, 1.0037, 0.6109,0.6109, 0.7175, 0.7108, 0.1288, 0.1289, 0.1301, 0.1301, 0.3659, 0.3659,0.3439, 0.3441.
 76. The digital communication system of claim 65,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 3.8751×10⁻² from thefollowing constellation points: −1.5003, −1.2326, −0.9288, −1.0219,−0.5860, −0.5860, −0.7430, −0.7215, −0.1015, −0.1016, −0.1366, −0.1366,−0.3901, −0.3901, −0.3167, −0.3170, 1.5005, 1.2327, 0.9289, 1.0219,0.5860, 0.5860, 0.7429, 0.7216, 0.1015, 0.1015, 0.1366, 0.1366, 0.3900,0.3900, 0.3167, 0.3169.
 77. The digital communication system of claim65, wherein the first PAM-32 symbol constellation achieves at least 90%of the gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis 1.6963×10⁻² from the following constellation points: −1.5003,−1.2326, −0.9288, −1.0219, −0.5860, −0.5860, −0.7430, −0.7215, −0.1015,−0.1016, −0.1366, −0.1366, −0.3901, −0.3901, −0.3167, −0.3170, 1.5005,1.2327, 0.9289, 1.0219, 0.5860, 0.5860, 0.7429, 0.7216, 0.1015, 0.1015,0.1366, 0.1366, 0.3900, 0.3900, 0.3167, 0.3169.
 78. The digitalcommunication system of claim 65, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.5003, −1.2326, −0.9288, −1.0219, −0.5860,−0.5860, −0.7430, −0.7215, −0.1015, −0.1016, −0.1366, −0.1366, −0.3901,−0.3901, −0.3167, −0.3170, 1.5005, 1.2327, 0.9289, 1.0219, 0.5860,0.5860, 0.7429, 0.7216, 0.1015, 0.1015, 0.1366, 0.1366, 0.3900, 0.3900,0.3167, 0.3169.
 79. The digital communication system of claim 65,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 2.9715×10⁻² from thefollowing constellation points: −1.4779, −1.2277, −0.9207, −1.0362,−0.5802, −0.5878, −0.7625, −0.7207, −0.0757, −0.0757, −0.1632, −0.1632,−0.4185, −0.4177, −0.3083, −0.3083, 1.4779, 1.2276, 0.9208, 1.0361,0.5802, 0.5878, 0.7625, 0.7207, 0.0758, 0.0758, 0.1632, 0.1632, 0.4184,0.4177, 0.3083, 0.3083.
 80. The digital communication system of claim65, wherein the first PAM-32 symbol constellation achieves at least 90%of the gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis 1.3058×10⁻² from the following constellation points: −1.4779,−1.2277, −0.9207, −1.0362, −0.5802, −0.5878, −0.7625, −0.7207, −0.0757,−0.0757, −0.1632, −0.1632, −0.4185, −0.4177, −0.3083, −0.3083, 1.4779,1.2276, 0.9208, 1.0361, 0.5802, 0.5878, 0.7625, 0.7207, 0.0758, 0.0758,0.1632, 0.1632, 0.4184, 0.4177, 0.3083, 0.3083.
 81. The digitalcommunication system of claim 65, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.4779, −1.2277, −0.9207, −1.0362, −0.5802,−0.5878, −0.7625, −0.7207, −0.0757, −0.0757, −0.1632, −0.1632, −0.4185,−0.4177, −0.3083, −0.3083, 1.4779, 1.2276, 0.9208, 1.0361, 0.5802,0.5878, 0.7625, 0.7207, 0.0758, 0.0758, 0.1632, 0.1632, 0.4184, 0.4177,0.3083, 0.3083.
 82. The digital communication system of claim 65,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 2.8949×10⁻² from thefollowing constellation points: −1.4700, −1.2271, −0.9199, −1.0418,−0.5753, −0.5899, −0.7720, −0.7166, −0.0710, −0.0710, −0.1686, −0.1686,−0.4245, −0.4221, −0.3073, −0.3074, 1.4677, 1.2269, 0.9198, 1.0416,0.5754, 0.5899, 0.7719, 0.7166, 0.0711, 0.0711, 0.1687, 0.1687, 0.4246,0.4222, 0.3074, 0.3075.
 83. The digital communication system of claim65, wherein the first PAM-32 symbol constellation achieves at least 90%of the gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis 1.1871×10⁻² from the following constellation points: −1.4700,−1.2271, −0.9199, −1.0418, −0.5753, −0.5899, −0.7720, −0.7166, −0.0710,−0.0710, −0.1686, −0.1686, −0.4245, −0.4221, −0.3073, −0.3074, 1.4677,1.2269, 0.9198, 1.0416, 0.5754, 0.5899, 0.7719, 0.7166, 0.0711, 0.0711,0.1687, 0.1687, 0.4246, 0.4222, 0.3074, 0.3075.
 84. The digitalcommunication system of claim 65, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.4700, −1.2271, −0.9199, −1.0418, −0.5753,−0.5899, −0.7720, −0.7166, −0.0710, −0.0710, −0.1686, −0.1686, −0.4245,−0.4221, −0.3073, −0.3074, 1.4677, 1.2269, 0.9198, 1.0416, 0.5754,0.5899, 0.7719, 0.7166, 0.0711, 0.0711, 0.1687, 0.1687, 0.4246, 0.4222,0.3074, 0.3075.
 85. The digital communication system of claim 65,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 2.0371×10⁻² from thefollowing constellation points: −1.4459, −1.2257, −0.9275, −1.0560,−0.5517, −0.6082, −0.8006, −0.7125, −0.0628, −0.0637, −0.1743, −0.1712,−0.4396, −0.4097, −0.2947, −0.3057, 1.4457, 1.2256, 0.9274, 1.0560,0.5516, 0.6082, 0.8005, 0.7125, 0.0629, 0.0638, 0.1744, 0.1713, 0.4396,0.4098, 0.2949, 0.3058.
 86. The digital communication system of claim65, wherein the first PAM-32 symbol constellation achieves at least 90%of the gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis 0.9726×10⁻² from the following constellation points: −1.4459,−1.2257, −0.9275, −1.0560, −0.5517, −0.6082, −0.8006, −0.7125, −0.0628,−0.0637, −0.1743, −0.1712, −0.4396, −0.4097, −0.2947, −0.3057, 1.4457,1.2256, 0.9274, 1.0560, 0.5516, 0.6082, 0.8005, 0.7125, 0.0629, 0.0638,0.1744, 0.1713, 0.4396, 0.4098, 0.2949, 0.3058.
 87. The digitalcommunication system of claim 65, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.4459, −1.2257, −0.9275, −1.0560, −0.5517,−0.6082, −0.8006, −0.7125, −0.0628, −0.0637, −0.1743, −0.1712, −0.4396,−0.4097, −0.2947, −0.3057, 1.4457, 1.2256, 0.9274, 1.0560, 0.5516,0.6082, 0.8005, 0.7125, 0.0629, 0.0638, 0.1744, 0.1713, 0.4396, 0.4098,0.2949, 0.3058.
 88. The digital communication system of claim 65,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 1.6887×10⁻² from thefollowing constellation points: −1.4100, −1.2114, −0.9340, −1.0567,−0.5593, −0.6315, −0.8180, −0.7261, −0.0437, −0.0791, −0.1969, −0.1577,−0.4687, −0.4099, −0.2822, −0.3293, 1.4100, 1.2114, 0.9340, 1.0567,0.5593, 0.6315, 0.8180, 0.7261, 0.0436, 0.0791, 0.1969, 0.1577, 0.4687,0.4099, 0.2822, 0.3292.
 89. The digital communication system of claim65, wherein the first PAM-32 symbol constellation achieves at least 90%of the gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis 0.7390×10⁻² from the following constellation points: −1.4100,−1.2114, −0.9340, −1.0567, −0.5593, −0.6315, −0.8180, −0.7261, −0.0437,−0.0791, −0.1969, −0.1577, −0.4687, −0.4099, −0.2822, −0.3293, 1.4100,1.2114, 0.9340, 1.0567, 0.5593, 0.6315, 0.8180, 0.7261, 0.0436, 0.0791,0.1969, 0.1577, 0.4687, 0.4099, 0.2822, 0.3292.
 90. The digitalcommunication system of claim 65, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.4100, −1.2114, −0.9340, −1.0567, −0.5593,−0.6315, −0.8180, −0.7261, −0.0437, −0.0791, −0.1969, −0.1577, −0.4687,−0.4099, −0.2822, −0.3293, 1.4100, 1.2114, 0.9340, 1.0567, 0.5593,0.6315, 0.8180, 0.7261, 0.0436, 0.0791, 0.1969, 0.1577, 0.4687, 0.4099,0.2822, 0.3292.
 91. The digital communication system of claim 65,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 1.0837×10⁻² from thefollowing constellation points: −1.3701, −1.1930, −0.9383, −1.0532,−0.5751, −0.6518, −0.8316, −0.7407, −0.0357, −0.0916, −0.2175, −0.1599,−0.4931, −0.4252, −0.2903, −0.3516, 1.3701, 1.1930, 0.9383, 1.0532,0.5751, 0.6518, 0.8316, 0.7407, 0.0357, 0.0916, 0.2175, 0.1599, 0.4931,0.4252, 0.2903, 0.3516.
 92. The digital communication system of claim65, wherein the first PAM-32 symbol constellation achieves at least 90%of the gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis 0.4672×10⁻² from the following constellation points: −1.3701,−1.1930, −0.9383, −1.0532, −0.5751, −0.6518, −0.8316, −0.7407, −0.0357,−0.0916, −0.2175, −0.1599, −0.4931, −0.4252, −0.2903, −0.3516, 1.3701,1.1930, 0.9383, 1.0532, 0.5751, 0.6518, 0.8316, 0.7407, 0.0357, 0.0916,0.2175, 0.1599, 0.4931, 0.4252, 0.2903, 0.3516.
 93. The digitalcommunication system of claim 65, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.3701, −1.1930, −0.9383, −1.0532, −0.5751,−0.6518, −0.8316, −0.7407, −0.0357, −0.0916, −0.2175, −0.1599, −0.4931,−0.4252, −0.2903, −0.3516, 1.3701, 1.1930, 0.9383, 1.0532, 0.5751,0.6518, 0.8316, 0.7407, 0.0357, 0.0916, 0.2175, 0.1599, 0.4931, 0.4252,0.2903, 0.3516.
 94. The digital communication system of claim 65,wherein the first PAM-32 symbol constellation comprises a set ofconstellation points where each point is within 0.7084×10⁻² from thefollowing constellation points: −1.3295, −1.1740, −0.9426, −1.0490,−0.5927, −0.6709, −0.8440, −0.7558, −0.0342, −0.0985, −0.2317, −0.1663,−0.5143, −0.4424, −0.3021, −0.3699, 1.3295, 1.1739, 0.9426, 1.0490,0.5927, 0.6709, 0.8440, 0.7558, 0.0342, 0.0985, 0.2317, 0.1663, 0.5144,0.4424, 0.3021, 0.3664.
 95. The digital communication system of claim65, wherein the first PAM-32 symbol constellation achieves at least 90%of the gain in parallel decode capacity of the given parallel decodecapacity achieved by a non-uniform PAM-32 symbol constellation optimizedfor parallel decode capacity at the given SNR relative to a uniformPAM-32 symbol constellation, and wherein the first PAM-32 symbolconstellation comprises a set of constellation points where each pointis 0.3063×10⁻² from the following constellation points: −1.3295,−1.1740, −0.9426, −1.0490, −0.5927, −0.6709, −0.8440, −0.7558, −0.0342,−0.0985, −0.2317, −0.1663, −0.5143, −0.4424, −0.3021, −0.3699, 1.3295,1.1739, 0.9426, 1.0490, 0.5927, 0.6709, 0.8440, 0.7558, 0.0342, 0.0985,0.2317, 0.1663, 0.5144, 0.4424, 0.3021, 0.3664.
 96. The digitalcommunication system of claim 65, wherein the first PAM-32 symbolconstellation achieves 100% of the gain in parallel decode capacity ofthe given parallel decode capacity achieved by a non-uniform PAM-32symbol constellation optimized for parallel decode capacity at the givenSNR relative to a uniform PAM-32 symbol constellation, and wherein thefirst PAM-32 symbol constellation comprises the following set ofconstellation points: −1.3295, −1.1740, −0.9426, −1.0490, −0.5927,−0.6709, −0.8440, −0.7558, −0.0342, −0.0985, −0.2317, −0.1663, −0.5143,−0.4424, −0.3021, −0.3699, 1.3295, 1.1739, 0.9426, 1.0490, 0.5927,0.6709, 0.8440, 0.7558, 0.0342, 0.0985, 0.2317, 0.1663, 0.5144, 0.4424,0.3021, 0.3664.